Figure 1.1: Man-in-the-middle attack on a public-key encryption scheme.
attain the desired level of security against such a powerful adversary? It is these types of questions
that are addressed by this dissertation.
We provide a concrete example to motivate this line of research. Consider a scenario in which
a client transmits a 48-bit credit-card number b
1
◦ ◦ b
48
to a server, perhaps along with a PIN
and whatever additional information is necessary to authorize a transaction. To ensure the client’s
privacy, all information sent from the client to the server is encrypted using the public key of the
server. Assume encryption is done bit-wise; i.e., a string is encrypted by encrypting each bit of the
string and then concatenating the resulting ciphertexts.
If the encryption algorithm c used to encrypt each bit of the string is secure against passive
eavesdroppers, the encryption of any polynomial-length string (as above) is also secure against a
passive eavesdropper. Unfortunately, the scheme is completely vulnerable against an active adversary
as shown by the following attack (cf. Figure 1.1). The adversary, upon observing transmission of
C
1
◦ ◦ C
48
(where C
i
represents encryption of b
i
), generates ciphertext C
′
by encrypting 0 using
the public-key of the server (note that the adversary has access to the server’s public key since
it is public information). The adversary then sends C
′
◦ C
2
◦ ◦ C
48
to the server and waits to
see whether the server accepts (i.e., authorizes the transaction) or not. If the server accepts, the
adversary concludes that the ﬁrst bit of the client’s credit-card number is 0; if the server rejects,
the ﬁrst bit of the credit-card number must be 1. Repeating this attack for each bit of the message,
the adversary obtains the client’s credit-card number after sending only 48 messages to the server!
This may be compared with randomly guessing the credit-card number, where the adversary needs
to send (on average) 2
47
messages to the server before learning the correct value. Clearly, man-in-
3
the-middle attacks such as this represent a real threat and need to be taken into account explicitly
when designing real-world systems.
Encryption is typically a simple, one-round protocol in which a single message is sent between
two parties. Dealing with man-in-the-middle attacks is even more challenging [44] when considering
more complex protocols consisting of many rounds of interaction, possibly among more than two
parties. Even the correct formalization of security against man-in-the-middle attacks in a way which
prevents undesired adversarial behavior while making realistic assumptions about the adversary’s
capabilities is non-trivial; in fact, entirely satisfactory deﬁnitions have not yet been given for some
cryptographic tasks. Protocol design has been even more diﬃcult, with few eﬃcient protocols known
that prevent man-in-the-middle attacks in a provably-secure way.
In practice, man-in-the-middle attacks are often dealt with by designing protocols that protect
against a list of known attacks; such an approach, however, leaves the protocol vulnerable to new
attacks as they are developed. Furthermore, many widely deployed protocols have only heuristic
arguments in favor of their security. Such an approach does not engender much conﬁdence, and,
unfortunately, many of these protocols have been broken soon after their introduction. A step in
the right direction is to “validate” a protocol by proving the protocol secure in an idealized model
such as the random oracle model
1
[12]. Clearly, however, proofs of security requiring no unrealistic
assumptions are preferable.
1.1 Summary of Contributions
In this dissertation, we consider the security of a wide range of cryptographic primitives against
man-in-the-middle attacks.
2
We present formal deﬁnitions of security against these attacks (in some
cases, these are the ﬁrst such deﬁnitions to appear) and give constructions of eﬃcient protocols
which are proven secure using standard cryptographic assumptions. Our contributions include the
following (terms below are deﬁned in Chapter 2 and the relevant chapter in which the term appears):
• In Chapter 3, we consider the problem of password-only authenticated key exchange over a
network completely controlled by an adversary. Here, the password shared by two parties is
explicitly modeled as a weak, human-memorizable secret which may be easily guessed by the
adversary [11, 24]. Due to the inherent weakness of the password, a secure protocol must
ensure (among other things) that an adversary can not determine the password — and hence
1
In the random oracle model, all participants are given oracle access to a truly random function. In practice, the
random oracle is instantiated with a cryptographic hash function. However, there are protocols which are secure in
the random oracle model but are known to be insecure when instantiated with any concrete function [29].
2
“Man-in-the-middle attack” is a broad term for any attack in which communication between honest parties may
be corrupted by an adversary. In this work we include precise deﬁnitions of security against man-in-the-middle attacks
for all primitives we consider.
4
break the protocol — using an oﬀ-line dictionary attack. We work in a setting in which a
common string is known to all parties [20] and users otherwise share only a weak password (in
particular, a public-key infrastructure is not required), and give the ﬁrst eﬃcient and provably-
secure protocol for password-only authenticated key exchange in this setting. A preliminary
version of this work has appeared previously [83].
• In Chapter 4, we consider the important cryptographic primitive of commitment [96]. There,
we describe the ﬁrst eﬃcient and non-malleable protocols for non-interactive, perfect commit-
ment. The security of our schemes may be based on either the RSA assumption or the dis-
crete logarithm assumption. We also propose and analyze a construction of a non-interactive,
non-malleable, standard commitment scheme which has near-optimal commitment length. A
preliminary version of this work has appeared previously [41].
• We next consider the case of interactive proofs of knowledge. Extending previous deﬁnitions
in the non-interactive setting [44, 110, 38], we formally deﬁne the notion of a non-malleable,
interactive proof of plaintext knowledge (PPK). We then give eﬃcient constructions of non-
malleable PPKs (for a number of standard cryptosystems) which remain secure even when
executed in a concurrent fashion. Finally, we show applications of these protocols to (1)
chosen-ciphertext-secure public-key encryption [99, 105], (2) password-based authenticated
key exchange in the public-key model [76, 22], (3) deniable authentication [44, 49], and (4)
identiﬁcation [56]. In many cases, our work provides the ﬁrst eﬃcient solutions to these prob-
lems based on factoring or other number-theoretic assumptions. These results are described
in Chapter 5.
Chapter 2 contains a description of the basic adversarial model considered in this dissertation
along with an overview of some previous deﬁnitional approaches to security against man-in-the-
middle attacks. There, we also summarize related work in this area and provide relevant crypto-
graphic background.
5
Chapter 2
The Model and Deﬁnitions
This chapter contains a description of the adversarial model
1
considered in this work, beginning
with a high-level overview and followed by a more detailed and formal deﬁnition. We also review
prior approaches to security against man-in-the-middle attacks and discuss relevant previous work
in this area. We conclude by introducing some notation and by deﬁning cryptographic terms used
throughout the remainder of this dissertation.
2.1 The Model: Overview
Our model includes some number of honest users (also called participants or parties) who may, if
and when they choose, take part in an execution of the protocol. The entire protocol is viewed
as taking place in two, distinct phases. In the initialization phase, the set of users is established;
furthermore, during this phase certain information may be generated and distributed among these
users. Following this, the protocol execution phase begins and the desired protocol is run. It is
important to note that the protocol may be executed an arbitrary (polynomial) number of times
during the protocol execution phase following only a single execution of the initialization phase.
During the protocol execution phase, each participant executes a local algorithm to generate
outgoing messages as speciﬁed by the protocol; these messages may depend on information received
in the initialization phase and on the set of messages sent and received by the participant thus
far. Delivery of these messages is, however, not guaranteed. In fact, as we will see below, all
communication between participants is completely controlled by an adversary. Details follow.
1
Although all our results may be cast in this model, we present a simpliﬁed view of the model when considering a
speciﬁc cryptographic task. Furthermore, we do not state a generic deﬁnition of security in this chapter; instead, we
give formal security deﬁnitions for each task we consider in the relevant chapter.
6
2.1.1 Initialization Phase
During the initialization phase, information is distributed among the parties using some (unspeciﬁed)
mechanism which is independent of the protocol itself. Examples of information which may be
distributed during this phase include:
• Secret information shared between two (or more) parties. Because the information is assumed
to be shared secretly during the initialization phase, this data is not available to the adversary.
We distinguish two types of secret information. A key always refers to a cryptographically-
strong key; i.e., a string with suﬃciently-large entropy so as to be resistant to guessing attacks
by the adversary. On the other hand, a password refers to a short string which may be easily
memorized by a human user (cryptographic keys, including public/secret keys, are too long to
be easily memorized!). Nothing about the entropy of a password is assumed. In particular,
passwords may be easily guessed by an adversary; hence, proofs of security (when passwords
are used) must explicitly take this fact into account.
• Public keys established as part of a public-key infrastructure (PKI). In this case, each par-
ticipant may generate a public key/secret key pair; the public key is distributed to all other
parties while the secret key remains known only to the party that generated it. Since the
public keys are freely distributed (with no eﬀort made to limit their dissemination), they are
also available to the adversary; of course, the corresponding secret keys are unavailable to the
adversary.
• Public information known to all parties [20]. A secure PKI has been notoriously diﬃcult to
implement and PKIs have well-known problems associated with registration, revocation, etc.
Much simpler than establishing a PKI is to ﬁx public parameters (generated by a known,
probabilistic algorithm) and distribute this information to all participants. An example of
this is the common random string model in which all participants possess identical copies of a
uniformly-distributed string. Since this information is publicly distributed, it is also assumed
to be available to the adversary.
Of course, the initialization phase may consist of some combination of the above; it is also possible
to consider a null initialization phase in which no information is shared in advance of protocol
execution.
It is assumed that the initialization phase speciﬁed for a particular protocol is executed cor-
rectly before protocol execution begins. Such an approach allows separation of the analysis of the
initialization phase from the analysis of the protocol itself. On the other hand, it is important to
7
recognize that this (in some sense) introduces a new assumption in the proofs of security. For this
reason, protocols using weaker (i.e., easier to achieve) initialization phases are preferable to protocols
requiring stronger set-up assumptions.
2.1.2 The Adversary
We consider a very powerful adversary who controls all communication between the honest partic-
ipants. More precisely, we view transmission of a message msg from A to B as a two-step process
in which A ﬁrst sends msg to the adversary and the adversary then sends msg to B. However,
the adversary may choose not to deliver messages (without notifying any participants), may deliver
messages out of order, may insert his own messages, and may arbitrarily modify messages in tran-
sit. Furthermore, the adversary may read all messages sent between users. This type of adversary,
though strong, is realistic in settings such as wireless communication networks in which messages
may be easily intercepted and potentially changed.
A protocol deﬁnes a probabilistic algorithm for each honest participant. The adversary is aware
of the algorithms deﬁning the protocol, and can cause participants to execute the protocol by
interacting with the participants in the appropriate way. A formal description of the adversary
appears in Section 2.2.
This adversarial model considered here is distinct from (and incomparable with) other models
which have been proposed for security in a multi-party setting (e.g., [66, 95, 61, 26]). In particu-
lar, other models typically assume authenticated channels (so that the identity of the sender of a
message is unambiguous) and guaranteed delivery of messages; neither condition is assumed here.
Additionally, other models typically allow some fraction of the participants to be corrupted by the
adversary in which case the adversary learns all their local information including long-term secrets
as well as state information used during protocol execution. Here, we generally assume that par-
ticipants themselves may not be corrupted (although in Chapter 3 a limited form of corruption is
considered).
2
Recently, a model capturing aspects of both the previous models and the model outlined here
has been proposed by Canetti [27]. Canetti has shown [27] that protocols secure in this model enjoy
very strong composability properties; in particular, protocols secure in this model are secure against
many types of man-in-the-middle attacks. The model is suﬃciently general to allow the formulation
of many cryptographic tasks; on the other hand, the resulting deﬁnitions are often complex and
protocols proven secure in this model are extremely ineﬃcient. For this reason, it is still often
2
In other models, the corrupted players may deviate from a correct execution of the protocol. Here — since
authenticated communication is not assumed — the adversary may send whatever messages he likes (claiming they
were sent by one of the participants) and in this way achieve the same eﬀect.
8
desirable to introduce alternate, simpler deﬁnitions for speciﬁc functionalities. We also remark that
Canetti’s model does not necessarily deal with all possible types of man-in-the-middle attacks (e.g.,
those considered in [81, 32, 44, 82]).
2.2 The Model: Details
The number of participants n is ﬁxed during the initialization phase and is polynomial in the
security parameter. We model the participants and the adversary as interactive Turing machines
(our deﬁnition follows [70, 62]):
Deﬁnition 2.1 A probabilistic, multi-tape Turing machine M is an interactive Turing machine (ITM)
if it satisﬁes the following:
• M’s tapes consist of: (1) a read-only input tape, (2) a read-only private-auxiliary-input tape,
(3) a read-only public-auxiliary-input tape, (4) a write-only output tape, (5) a read-and-write
work tape, (6) a read-only random tape, (7) a read-only communication-in tape, and (8) some
number of write-only communication-out tapes.
• M is message-driven; in other words, M is activated when a new message is written on its
communication-in tape and deactivated when it writes a special symbol ⊥ on its communication-
out tape. A period from when M is activated to when it is next deactivated is called a period
of activation. We may analogously deﬁne a period of deactivation.
• The content written on the communication-out tape during a particular period of activation
(not including the symbol ⊥) is called the message sent by M during that period; the content
written on the communication-in tape during a particular period of deactivation is called the
message received by M during that period.
To fully deﬁne our model, we describe how the participants jointly execute a protocol in the presence
of an adversary. We stress that joint execution among n participants does not mean that they must
all take part in every invocation of the protocol; it simply means that each participant has the option
of executing the protocol with some other (subset of the) participants of its choosing.
Deﬁnition 2.2 Let M
1
, . . . , M
n
, and / be interactive Turing machines. We say that M
1
, . . . , M
n
are linked via / if:
• Each M
i
has a single communication-out tape.
• / has n communication-out tapes labeled 1, . . . , n.
9
• The public-auxiliary-input tapes of M
1
, . . . , M
n
, and / coincide.
• The i
th
communication-out tape of / coincides with the communication-in tape of M
i
. The
communication-out tapes of each M
i
coincide with the communication-in tape of /.
• All other tapes are distinct.
M
1
, . . . , M
n
are called the participants and / is called the adversary.
The system is initialized as follows: the random tapes of each ITM are chosen independently at
random. The work tape, input tape, output tape, communication-in tape, and communication-out
tape(s) of all ITMs are initially empty. The public-auxiliary-input tape contains 1
k
so that the
security parameter is well-deﬁned for all ITMs. The public- and private-auxiliary-input tapes may
contain additional information generated during the initialization phase
3
(as described in Section
2.1.1). Typically, this auxiliary information will be the output of some probabilistic initialization
protocol. As an illustrative example, an ℓ-bit secret may be shared between users i and j by choosing
w at random from¦0, 1¦
ℓ
and writing w on the private auxiliary-input tapes of M
i
and M
j
(and on no
other tapes). Public-key generation by player i is achieved by running a key generation algorithm
to obtain (SK, PK), having (i, PK) written on the public auxiliary-input tape, and having SK
written on the private auxiliary-input tape of M
i
. Finally, public information may be distributed by
running some algorithm to yield σ and then writing σ on the public auxiliary-input tape. Note that
/ has access to the information written on the public auxiliary-input tape; this explicitly models
the public nature of this information.
During the protocol execution phase, / is the ﬁrst to be activated. Computation proceeds as
follows: / begins by writing a (possibly empty) message on its i
th
communication-out tape, for some
i. When / is done, M
i
is activated and may read the message written on its communication-in tape;
M
i
also writes a (possibly empty) message on its communication-out tape. We assume the intended
recipient is clear from a description of the protocol; alternately, one may modify any protocol so that
M
i
sends message (i, j, msg) whenever it would otherwise send msg intended for player j. When M
i
is done, the adversary is re-activated. Even though the adversary has only a single communication-
in tape, / can determine the sender of any message because / decides the order of activation. /
may write any message of his choice on the communication-in tape of any recipient; we stress that
the adversary is not required to deliver those messages output by the honest parties. Computation
proceeds in this manner until the adversary halts.
3
In the non-uniform model, additional private auxiliary input may be given to the adversary. In this case, it is
often crucial for the security of the protocol that this auxiliary information be ﬁxed before the initialization phase.
Alternately, this auxiliary information may arise from the composition of two (or more) protocols.
10
All participants are assumed to run in polynomial time (in the security parameter k). In partic-
ular, there exists some polynomial p() such that each M
i
halts within at most p(k) steps regardless
of the behavior of the adversary. / is also assumed to run in polynomial time.
2.3 Security Against Man-in-the-Middle Attacks
The distinguishing feature of the above model is that there is no direct communication between any
of the honest parties. Instead, all communication is “routed” through /. Yet, not all hope is lost in
the face of such a strong adversary. Consider the problem of security against passive eavesdropping.
4
If a PKI is established during the initialization phase, the desired level of security may be achieved
by having A encrypt all messages (using the public key of B) before sending them to B.
It is instructive to consider other types of attacks, beyond mere eavesdropping, so as to recognize
what is not achieved in this example. First of all, there is no guarantee that messages from A will
ever be received by B. Worse, however, is the fact that B is not assured that messages he receives
are actually from A, since the adversary (who has access to PK
B
) can also encrypt messages and
send them to B. In general, these types of attacks cannot be prevented (although there may be
ways to ensure that these attacks are detected) due to the strong adversarial model we consider; for
example, we do not assume authenticated channels between parties.
This discussion indicates that certain things cannot be achieved in our model. More precisely,
we can not hope to prevent the adversary from:
• Preventing communication between parties of his choice.
• Attempting to impersonate one of the honest participants.
• Faithfully forwarding messages, thereby “copying” all messages sent by a particular user (we
will return to this point below).
Our goal, then, is to prevent more devious attacks whose feasibility is not an immediate consequence
of the model. As an example of such an attack, assume A is sending a “yes/no” response (represented
by a “1/0”) to B. The adversary might be able to “ﬂip” the contents of A’s message by somehow
modifying the ciphertext C sent by A. Note that the adversary may (in theory) achieve this without
ever determining A’s original message, and thus the privacy of the encryption scheme is not violated.
Preventing this type of attack is among the problems considered in this work.
The above discussion suggests that deﬁning security against man-in-the-middle attacks is not
simple. Indeed, many cryptographic primitives still have no universally agreed-upon deﬁnitions for
4
In the context of our model, we may deﬁne a passive eavesdropper as an adversary who always reliably forwards
messages, without any modiﬁcations, to the intended recipient.
11
security against man-in-the-middle attacks. Yet, it will be useful to review some deﬁnitions which
have been suggested previously (in a variety of diﬀerent settings) for eventual comparison with the
deﬁnitions under which we prove security of our constructions.
2.3.1 Deﬁnitional Approaches
Ping-pong protocols and early approaches. Man-in-the-middle attacks on cryptographic pro-
tocols have long been recognized as a fundamental problem. Early attempts to deal with such attacks
focused on the security of ping-pong protocols (in which the output of a party is a simple function
of the current input) against adversarial man-in-the-middle behavior [46, 45, 119, 52]. Although a
formal approach is taken, certain limitations of this approach are apparent. First is that the class of
ping-pong protocols is very limited; in particular, it does not include protocols which maintain state
during a multi-round execution. Thus, the approach does not address some very natural scenarios;
e.g., a party who executes a protocol only once and then refuses to execute the protocol again. Sec-
ond, the analysis of ping-pong protocols assumes that cryptographic primitives such as encryption
are ideal; for example, the analysis assumes that given a ciphertext C which is an encryption of some
message m, it is infeasible to come up with a diﬀerent ciphertext C
′
which is also an encryption of
m. Real-life encryption protocols, however, are not ideal; the resulting ciphertexts are simply bit
strings which may be manipulated in a variety of ways (cf. the man-in-the-middle attack in Figure
1.1). Finally, secure ping-pong protocols have typically been designed assuming identities have been
established for all participants. In a large network, however, this may not be the case.
The work of Rivest and Shamir [106] represents another early attempt to prevent man-in-the-
middle attacks (in this case, for a key-exchange protocol). Although their approach is novel, they
neither deﬁne nor prove security of their protocol. Analysis of their protocol shows that users need
to (eﬀectively) share a cryptographic key in advance of protocol execution. Furthermore, although
the protocol achieves a “basic” level of security as outlined by the authors, the protocol is easily
seen to be susceptible to more complex attacks.
Non-malleability and simulatability. The limitations of the approaches mentioned above illus-
trate the need for formal approaches that do not idealize cryptographic primitives and that allow for
the analysis of more complex, “real-world” protocols. Dolev, Dwork, and Naor [44] were the ﬁrst to
present an approach that may be applied to man-in-the-middle attacks in many diﬀerent contexts.
Speciﬁcally, they introduce the notion of non-malleability and formally deﬁne non-malleability of
commitment, encryption, and (interactive) zero-knowledge proofs. Their approach may be viewed
as follows. Assume participants A and B, jointly executing some protocol, are linked via adversary
/. The protocol is said to be non-malleable if (informally) B’s “view” in the real world — in which
12
B interacts with man-in-the-middle / — can be simulated by an eﬃcient algorithm that does not
interact with A. As an informal example, consider the case of public-key encryption. Assume that
A encrypts a message m (chosen from some speciﬁed distribution T) to yield a ciphertext C. Let C
′
be a second ciphertext generated by / after receiving C. Note that B can decrypt C
′
to obtain some
message m
′
, and this real-world experiment therefore deﬁnes some distribution T
′
over (m, m
′
). Very
loosely speaking, a non-malleable encryption scheme has the property that there exists a simulator
(who is not given any ciphertext) that can output a ciphertext C
′′
(with decryption m
′′
) such that
the distribution T
′′
over (m, m
′′
) (where m is chosen from T) is indistinguishable from distribution
T
′
. In other words, ciphertext C is of no help to the adversary in constructing C
′
.
There is one additional point, however, which needs to be taken into account. / can always
exactly copy A’s messages and forward them to B; in the encryption example, the adversary can
output simply C
′
= C. On the other hand, a simulator cannot do the same. Thus, the formal
deﬁnition of non-malleability must rule out such behavior.
A related approach is to require that a real execution of the protocol be simulatable (in a way
made more precise below) by a simulator who is given access to an idealized functionality performing
the same task. Depending on the precise deﬁnition, simulatability may guarantee (some form of)
non-malleability. As an example, this approach has been used to deﬁne the security of key-exchange
protocols [4, 114, 24, 30] against man-in-the-middle attacks. Using this methodology, an ideal model
is deﬁned in which the desired task is carried out. For the case of key exchange, this idealized model
might include a special, trusted party who generates session keys and delivers these keys securely to
any pair of users upon request. A simulator’s interaction with this ideal model is limited to a speciﬁc
set of actions; for example, in the case of key exchange the simulator might be allowed to request
that session keys be established between any two parties of its choosing. During a real execution of
the protocol, of course, messages must be exchanged (via the adversary) between parties desiring to
establish a session key. A protocol is said to be secure if these messages (that is, a real execution of
the protocol) can be eﬃciently simulated — for an arbitrary adversary — by a simulator running
in the ideal model. This implies that anything that can be done by the adversary attacking the
real protocol can be done equally well by a simulator attacking the ideal protocol. Since the ideal
protocol is secure against man-in-the-middle attacks by deﬁnition (assuming the ideal functionality
is deﬁned properly), this implies that the real-world protocol is secure.
Recent work by Canetti [27] presents a uniﬁed framework in which to analyze protocols, roughly
along the lines sketched above (although we have omitted many important details in our discussion).
One of the contributions of this framework is that protocols simulatable under the given deﬁnition
are automatically secure against (certain classes of) man-in-the-middle attacks. Yet, it is diﬃcult
13
(and in certain cases, impossible [28]) to design secure protocols in this model without additional
assumptions. Furthermore, for speciﬁc tasks (i.e., key exchange) it is often beneﬁcial to design a
model with that task speciﬁcally in mind.
Oracle-based models. While the previously-mentioned approaches are appealing, they are often
very cumbersome to work with. Furthermore, they may in certain cases be too restrictive when
full simulatability is not required. This motivates other deﬁnitions which are easier to work with,
and — perhaps more importantly — often yield simpler and more eﬃcient protocols. One such
approach is to allow the adversary to interact with oracles speciﬁed in a manner appropriate for
the task at hand. For example, in the case of encryption, the adversary may be given access to a
decryption oracle which takes as input any ciphertext C and returns the underlying plaintext. We
can then deﬁne a new type of secure encryption as follows [99, 105, 6] (see also Deﬁnition 2.5): First,
the adversary receives ciphertext C. Then, the adversary may interact with the decryption oracle,
obtaining the plaintext corresponding to any ciphertext(s) C
′
of the adversary’s choosing.
5
The
encryption scheme is said to be “chosen-ciphertext secure” if the contents of C remain hidden from
the adversary even after interaction with the decryption oracle. Oracle-based models have also been
proposed for analyzing key exchange and mutual authentication protocols [13, 15, 11].
In some cases, an oracle-based deﬁnition of security has been proven equivalent to a simulation-
based deﬁnition [114] or to non-malleability [6, 16]. In these cases, it is often signiﬁcantly easier to
prove security of a protocol under the oracle-based deﬁnition; deﬁnitional equivalence then implies
that the protocol inherits the security properties guaranteed by the (seemingly) stronger deﬁnition.
Other approaches. Other approaches to man-in-the-middle attacks are also possible, especially
when the security desired is diﬀerent from the security guarantees outlined above. For example,
when user identities and a PKI are assumed, we may imagine a situation in which a zero-knowledge
proof should be “meaningful” only to a speciﬁc receiver [81, 32]. Here, even if the adversary copies
a proof to a third party, the third party should not be convinced by the proof; thus, the deﬁnition
of security must deal explicitly with copying instead of simply disallowing it. One may also consider
the complementary notion in which proofs must remain uniquely identiﬁed with a particular prover
[19, 44, 82]; in this case, even when an adversary copies a proof it should remain clear which party
actually generated it.
5
As before, we need to explicitly rule out copying. Thus, the adversary is not allowed to submit C
′
= C to the
decryption oracle.
14
2.4 Previous Work
Here, we survey some of the most important work dealing with non-malleability. More detailed
discussion of previous work related to a particular application appears in the relevant chapters of
this thesis.
As mentioned above, deﬁnitions for non-malleable encryption, commitment, and (interactive)
zero-knowledge proofs have previously appeared [44]. Constructions of these primitives which
are provably non-malleable are also known [44]. Subsequently, deﬁnitions for non-malleable, non-
interactive zero-knowledge proofs (and non-interactive proofs of knowledge) appeared [110, 38], along
with constructions achieving these deﬁnitions. These were also used to construct improved non-
malleable encryption protocols, following the paradigm established in [99]. A revised deﬁnition of
non-malleable commitment (appropriate for the case of perfect commitment) appears in [40, 58].
The ﬁrst construction of a non-interactive, non-malleable commitment scheme is given in [40].
The above constructions are all based on general assumptions and are therefore highly imprac-
tical. Only one eﬃcient and provably-secure construction of a non-malleable encryption scheme is
known [36].
6
An eﬃcient (interactive) non-malleable commitment scheme has been given [58]. In
the random oracle model, many eﬃcient chosen-ciphertext-secure encryption schemes are known;
e.g., [12, 14].
Relations among deﬁnitions of security for public-key encryption (especially non-malleability vs.
chosen-ciphertext security) are considered in [6, 16]. Similar relations have been established for
private-key encryption [84], where the adversary’s interaction with an encryption oracle must also
be taken into account. Other areas for which a formal approach to man-in-the-middle attacks has
been given include: key exchange and mutual authentication [18, 13, 15, 4, 76, 114, 22, 11, 24, 23, 92,
30, 65], deniable authentication [49, 50, 48], identiﬁcation [7], and designated-veriﬁer proofs [81, 32].
2.5 Notation and Preliminaries
2.5.1 Notation
We adopt the now-standard notation of Goldwasser, Micali, and Rackoﬀ [70]. The set of n-bit strings
is denoted by ¦0, 1¦
n
, the set of all binary strings of length at most n is denoted by ¦0, 1¦
≤n
, and the
set of all ﬁnite, binary strings is denoted by ¦0, 1¦
∗
. Concatenation of two strings x
1
, x
2
is denoted
by x
1
◦ x
2
or x
1
[ x
2
. The output y of a deterministic function f on input x
1
, . . . , x
n
is denoted by
y := f(x
1
, . . . , x
n
). Similarly, if random tape r is ﬁxed, the (deterministic) output y of probabilistic
algorithm A on input x
1
, . . . , x
n
and random tape r is denoted by y := A(x
1
, . . . , x
n
; r). We let
6
Very recently, other eﬃcient constructions of non-malleable encryption schemes have been given [37].
15
y ← A(x
1
, . . . , x
n
) refer to the (randomized) experiment in which r is chosen uniformly at random
and y is set to the output of A(x
1
, . . . , x
n
; r). For a ﬁnite set S, the notation x ← S means that x
is chosen uniformly at random from S. If p(x
1
, x
2
, . . .) is a predicate, the notation
Pr [x
1
← S; x
2
← A(x
1
, . . .); : p(x
1
, x
2
, . . .)]
denotes the probability that p(x
1
, x
2
, . . .) is true after ordered execution of the listed experiments.
An important convention is that all appearances of a given variable in a probabilistic statement
refer to the same random variable. If f is a deterministic function, the predicate f(x
1
, . . . , x
n
) = y
is true exactly when the output of f(x
1
, . . . , x
n
) is equal to y; for a randomized algorithm f, we
write (slightly abusing notation) f(x
1
, . . . , x
n
) = y to refer to the predicate f(x
1
, . . . , x
n
; r) = y for
randomly chosen r.
A probabilistic, polynomial-time (ppt) ITM M is one for which there exists a polynomial p()
such that, for all inputs x
1
, . . . , x
n
, all random tapes r, and arbitrary behavior of other machines with
which M is interacting, M(x
1
, . . . , x
n
; r) runs in time bounded by p([x
1
◦ ◦ x
n
[). An expected-
polynomial-time ITM M is one for which there exists a polynomial p() such that, for all inputs
x
1
, . . . , x
n
and regardless of the behavior of other machines with which M is interacting, the expected
running time of M(x
1
, . . . , x
n
; r) (over choice of r) is bounded by p([x
1
◦ ◦ x
n
[). All algorithms
we consider are (at least implicitly) given the security parameter k (in unary) as input; the lengths
of other inputs are always polynomially related to k and therefore, in most cases, running times
are measured as a function of k. All algorithms (including adversarial algorithms) are modeled as
uniform Turing machines (although our results extend to the non-uniform case by modifying the
computational assumptions appropriately).
A function ε : N →R
+
is negligible if for all c > 0 there exists an n
c
such that, for n > n
c
we have
ε(n) < 1/n
c
. In other words, ε() is asymptotically bounded from above by any inverse polynomial.
A function f : N →R
+
is non-negligible (or noticeable) if there exists a c > 0 and an n
c
such that,
for n > n
c
we have f(n) > 1/n
c
. Note that it is possible for a function to be neither negligible
nor non-negligible. Two distributions X
k
, X
′
k
(indexed by parameter k ∈ N) are computationally
indistinguishable if, for all ppt distinguishing algorithms D, the following is negligible (in k):
[Pr[x ← X
k
: D(x) = 1] −Pr[x ← X
′
k
: D(x) = 1][ .
Two distributions X
k
, X
′
k
are statistically indistinguishable if the following is negligible (in k):
¸
α
[Pr[x ← X
k
: x = α] −Pr[x ← X
′
k
: x = α][ .
In this case, we may also say that X
k
, X
′
k
have negligible statistical diﬀerence. Note that statistical
indistinguishability implies computational indistinguishability.
16
2.5.2 Cryptographic Assumptions
Hardness of factoring. This assumption, one of the most widely known, states that it is infea-
sible for any algorithm to ﬁnd the factors of a random product of two primes. More formally, let
2-factor
k
def
= ¦pq [ p and q are prime; [p[ = [q[ = k¦. Then, the factoring assumption states that
for every ppt algorithm A, the following probability is negligible (in k):
Pr[N ← 2-factor
k
; (p, q) ← A(N) : p q = N].
If we restrict ourselves to N of a special form the factoring assumption needs to be appropriately
modiﬁed. For example, N is a Blum integer if N = pq with p, q prime and p, q = 3 mod 4; it is
widely believed that factoring Blum integers is intractable.
A result due to Rabin [104] shows that the hardness of inverting the squaring function, deﬁned by
f(x) = x
2
mod N, is equivalent to the hardness of factoring. A similar result holds for the function
f
i
(x) = x
2
i
mod N (for ﬁxed i). Note this implies that both of these functions can be eﬃciently
inverted if the factorization of N is known.
The RSA assumption [107]. Informally, for a modulus N = pq which is the product of two
primes, a ﬁxed e which is relatively prime to ϕ(N), and a random r ∈ Z
∗
N
, the RSA assumption
states that it is infeasible to compute r
1/e
mod N. More formally, the RSA assumption states that
for all ppt algorithms A, the following probability is negligible (in k):
Pr [N ← 2-factor
k
; r ←Z
∗
N
; x ← A(N, e, r) : x
e
= r mod N] ,
where e is any number relatively prime to ϕ(N). If the factorization of N is known, then for all e
relatively prime to ϕ(N) and for all r ∈ Z
∗
N
, the value r
1/e
can be computed eﬃciently. Thus, the
RSA assumption is at least as strong as the assumption that factoring is hard.
Discrete-logarithm-based assumptions [42]. Assume a ﬁnite, cyclic group G such that the
order of G is prime (this condition is not essential for the assumptions below, yet all G used in this
work have this property). For any elements g, h ∈ G with g = 1, the value log
g
h is well-deﬁned
as the unique a ∈ Z
|G|
for which g
a
= h. For convenience, we use groups G deﬁned as follows: let
p = 2q + 1, with p, q prime and let G be the (unique) subgroup of Z
∗
p
with order q.
The discrete logarithm assumption states that, given randomly-chosen elements g, h, computing
log
g
h is infeasible. More formally, let ( be an eﬃcient algorithm which, on input 1
k
, generates p, q
prime with p = 2q + 1 and [q[ = k (thereby deﬁning group G). The discrete logarithm assumption
states that for all ppt algorithms A, the following is negligible (in k):
Pr

(p, q) ← ((1
k
); g, h ←G; a ← A(p, q, g, h) : g
a
= h

.
17
Related (but possibly stronger) assumptions include the computational Diﬃe-Hellman (CDH) and
decisional Diﬃe-Hellman (DDH) assumptions. For G deﬁned as above, the CDH assumption states
that for any ppt algorithm A, the following is negligible (in k):
Pr

(p, q) ← ((1
k
); g ←G; x, y ←Z
q
: A(p, q, g, g
x
, g
y
) = g
xy

.
The DDH assumption states that the following distributions are computationally indistinguishable,
where the ﬁrst distribution is over random tuples and the second is over Diﬃe-Hellman tuples:
¦(p, q) ← ((1
k
); g ←G; x, y, z ←Z
q
: (p, q, g, g
x
, g
y
, g
z
)¦
¦(p, q) ← ((1
k
); g ←G; x, y ←Z
q
: (p, q, g, g
x
, g
y
, g
xy
)¦.
Clearly, the DDH assumption implies the CDH assumption which in turn implies the discrete loga-
rithm assumption. It is not known whether the converses hold.
Expected-polynomial-time algorithms. Occasionally, we will need to assume that the above-
mentioned problems are hard even with respect to algorithms which are permitted to run in expected
polynomial time (the assumptions above are stated with respect to ppt algorithms). Although these
represent stronger assumptions, they are still widely believed to hold for the problems listed above.
2.5.3 Cryptographic Tools
Cryptographic hash functions. Two distinct notions of cryptographic hash functions are primar-
ily used.
7
The ﬁrst notion is that of (families of) collision-resistant hash functions (CRFs). These
are functions h for which it is infeasible to ﬁnd distinct pre-images x, x
′
such that h(x) = h(x
′
)
(typically, h compresses its input). Formally, let H = ¦H
k
¦
k∈N
, where each H
k
is a ﬁnite family
of eﬃciently-evaluable functions such that, for each h ∈ H
k
, we have h : ¦0, 1¦
∗
→ ¦0, 1¦
≤p(k)
for
some polynomial p(). Such a collection is called collision-resistant if for any ppt algorithm A, the
following is negligible (in k):
Pr[h ← H
k
; (x, x
′
) ← A(1
k
, h) : x = x
′
∧ h(x) = h(x
′
)].
Collision-resistant hash families are known to exist based on the hardness of factoring or the discrete
logarithm assumption; see [109] for minimal assumptions on which CRFs may be based.
In practice, very eﬃcient hash functions mapping ¦0, 1¦
∗
to a ﬁxed output length are used;
examples include SHA-1 and MD5. Although the security of these hash functions is at best heuristic,
these functions are generally considered to be collision-resistant for all practical purposes.
7
A third approach is to model speciﬁc cryptographic hash functions as random oracles; see Chapter 1, footnote 1.
As discussed there, this approach represents an idealized view of hash functions which cannot be realized in practice.
Since we do not use this approach in this thesis, we omit further discussion.
18
The second notion is that of (families of) universal one-way hash functions. Here, an adversary
ﬁrst outputs a value x before receiving a hash function h chosen at random from the family (recall
that for collision-resistant hash functions the adversary may choose both x and x
′
after receiving h).
It is then infeasible for the adversary to ﬁnd an x
′
= x such that h(x) = h(x
′
). More formally, let
H = ¦H
k
¦
k∈N
, where each H
k
is a ﬁnite family of eﬃciently-evaluable hash functions such that, for
each h ∈ H
k
, we have h : ¦0, 1¦
∗
→ ¦0, 1¦
≤p(k)
for some polynomial p(). Such a collection is called
universal one-way if for any ppt algorithm A = (A
1
, A
2
), the following is negligible (in k):
Pr[(x, s) ← A
1
(1
k
); h ← H
k
; x
′
← A
2
(1
k
, s, h) : x = x
′
∧ h(x) = h(x
′
)].
Universal one-way hash functions were introduced by Naor and Yung [98], who provide a construction
based on any one-way permutation. Subsequently, it was shown that one-way functions are suﬃcient
for the construction of universal one-way hash functions [108]. Note that any collision-resistant hash
family is also universal one-way; however, there is evidence that the existence of collision-resistant
hash functions is a strictly stronger assumption [85].
Public- and private-key encryption. An encryption scheme allows one party to send a message
to another such that the contents of the message remain hidden from anyone intercepting the com-
munication. Though the intuition is simple, formalizing this intuition correctly requires care. The
generally-accepted notion of security for encryption is semantic security, introduced by Goldwasser
and Micali [69]; this deﬁnition states (informally) that anything which can be eﬃciently computed
about a plaintext message when given access to the encryption of that message can be eﬃciently
computed without access to the encryption of the message (in particular, this implies that the mes-
sage itself cannot be determined without the decryption key). A second deﬁnition of security is
that of indistinguishability [69]; here, an adversary outputs two messages x
0
, x
1
and is then given an
encryption of one of them (chosen at random). The adversary succeeds if he can determine which
message was encrypted. An encryption scheme is indistinguishable if the success probability of any
ppt adversary is negligibly close to 1/2 (the adversary can always succeed half the time by guessing
randomly).
The basic deﬁnition of indistinguishability given above is equivalent to that of semantic security
[69]. Under the basic deﬁnition, however, the adversary is given only the ciphertext. Subsequent
work has considered stronger attacks in the public- [99, 105, 6] and symmetric-key [5, 84] settings.
We begin with a generic deﬁnition of an encryption scheme:
Deﬁnition 2.3 An encryption scheme Π is a triple of algorithms (/, c, T) such that, for some
polynomial p():
19
• The key generation algorithm / is a ppt algorithm that takes as input a security parameter k
(in unary) and returns keys sk and pk.
• The encryption algorithm c is a ppt algorithm that takes as input 1
k
, key pk, and a message
x ∈ ¦0, 1¦
≤p(k)
and returns a ciphertext C (we denote this by C ← c
pk
(x)).
• The decryption algorithm T is a deterministic, poly-time algorithm that takes as input 1
k
, key
sk, and a ciphertext C and returns either a message x ∈ ¦0, 1¦
≤p(k)
or a special symbol ⊥ to
indicate that the ciphertext C is invalid (we denote this by x := T
sk
(C)).
We require that for all k, for all (sk, pk) which can be output by /(1
k
), for all x ∈ ¦0, 1¦
≤p(k)
, and
for all C which can be output by c
pk
(x), we have T
sk
(C) = x.
We may deﬁne a private-key (also known as symmetric-key) encryption scheme as one in which, for
all (sk, pk) output by /(1
k
), we have pk = sk. A public-key encryption scheme will have pk = sk;
note that this is not implied by the deﬁnition above, yet will be implied by the deﬁnition of security
given below.
We ﬁrst present the basic notion of indistinguishability for public-key encryption [69]. Note that
the adversary is explicitly given the public key at all times.
Deﬁnition 2.4 Π = (/, c, T) is a public-key encryption scheme secure in the sense of indistinguisha-
bility (equivalently [69], semantically secure) if for any ppt adversary A = (A
1
, A
2
), the following is
negligible (in k):

.
One may also consider stronger classes of adversaries which are given access to certain oracles. For an
adversary attacking a public-key encryption scheme, the only oracle of interest is a decryption oracle
T
sk
() which returns the decryption of any ciphertext C given to it by the adversary. Encryption
schemes which remain secure even against this class of adversaries are termed (adaptive
8
) chosen-
ciphertext secure [105].
Deﬁnition 2.5 Π = (/, c, T) is a chosen-ciphertext-secure (CCA2) public-key encryption scheme if
for any ppt adversary A = (A
1
, A
2
), the following is negligible (in k):

,
8
A deﬁnition of non-adaptive chosen-ciphertext security, in which A
1
has access to the oracle but A
2
does not, is
also possible. In fact, this was the ﬁrst such deﬁnition presented [99]. However, since we do not explicitly use this
weaker deﬁnition in this work, we omit further discussion.
20
where we require that A
2
not submit C to its decryption oracle.
Semantically-secure public-key encryption schemes may be based on any (family of) trapdoor
functions with polynomial preimage-size [9]; in particular, public-key cryptosystems exist based on
the hardness of factoring [104] and RSA [107] assumptions. Security of the El-Gamal encryption
scheme [51] is based on the DDH assumption. CCA2 public-key encryption schemes may be based
on trapdoor permutations [44]; an eﬃcient construction based on the DDH assumption is also known
[36].
Semantic security for private-key encryption is deﬁned in an analogous manner. Here, one may
also consider adversaries with access to an encryption oracle [84]; however, we will not need such
a notion for the present work. For completeness, we provide the deﬁnition for (adaptive) chosen-
ciphertext security here (we omit pk since, for private-key encryption, pk = sk).
Deﬁnition 2.6 Π = (/, c, T) is a chosen-ciphertext-secure private-key encryption scheme if for any
ppt adversary A = (A
1
, A
2
), the following is negligible (in k):

,
where we require that A
2
not submit C to its decryption oracle.
One-way functions are necessary and suﬃcient for constructing semantically-secure [78] and chosen-
ciphertext-secure [44] private-key encryption schemes.
Message authentication codes. A message authentication code (mac) allows two parties, who
have shared a secret key in advance, to authenticate their subsequent communication. More formally,
a mac is a key-based algorithm which associates a tag with every valid message. The tag for
a particular message may be eﬃciently veriﬁed by the party sharing the key. Furthermore, an
adversary who sees many message/tag pairs is unable to forge a tag on a new message. We begin
with a deﬁnition of a mac algorithm and then deﬁne an appropriate notion of security.
Deﬁnition 2.7 A message authentication code Π is a triple of algorithms (/, mac, Vrfy) such that,
for some polynomial p():
• The key generation algorithm / is a ppt algorithm that takes as input a security parameter k
(in unary) and returns key sk.
• The tagging algorithm mac is a ppt algorithm that takes as input 1
k
, a key sk, and a message
m ∈ ¦0, 1¦
≤p(k)
and returns a tag T (we denote this by T ← mac
sk
(m)).
21
• The veriﬁcation algorithm Vrfy is a deterministic algorithm that takes as input 1
k
, a key
sk, a message m ∈ ¦0, 1¦
≤p(k)
, and a tag T and returns a single bit (we denote this by
b := Vrfy
sk
(m, T)).
We require that for all k, all sk output by /(1
k
), all m ∈ ¦0, 1¦
≤p(k)
, and all T output by mac
sk
(m),
we have Vrfy
sk
(m, T) = 1.
A mac is secure if an adversary is unable to forge a valid message/tag pair. Yet we need to specify
the class of adversary we consider (i.e., the type of attack allowed). Following [10], we consider the
strongest type of attack: the adversary may interact — adaptively and polynomially-many times —
with an oracle mac
sk
() that returns the correct tag for any message submitted by the adversary.
Deﬁnition 2.8 A message authentication code Π = (/, mac, Vrfy) is secure under adaptive chosen
message attack if for all ppt forging algorithms T, the following probability is negligible (in k):
Pr[sk ← /(1
k
); (m, T) ← T
mac
sk
(·)
(1
k
) : Vrfy
sk
(m, T) = 1],
where we require that T not be a tag previously output by mac
sk
() on input m.
Since this deﬁnition is now standard, the term “secure mac” in this work refers to a mac that is
secure under adaptive chosen message attack. A variety of mac constructions are known, and a
secure mac may be based on any one-way function [63, 77].
Signatures. A formal deﬁnition of security for signature schemes was ﬁrst given by Goldwasser,
Micali, and Rivest [71]. Here, a signer publishes a (public) veriﬁcation key V K and keeps secret a
signing key SK. A signing algorithm, which takes as additional input a signing key SK, associates
a signature with every valid message; this signature may be validated by anyone who knows the
corresponding veriﬁcation key. As with a secure mac, an adversary should be unable to forge a
valid signature on a previously-unsigned message. The formal deﬁnitions of a signature scheme and
its security under adaptive chosen-message attack exactly parallel those given above for macs. For
completeness, we give the full deﬁnitions here (following [71]).
Deﬁnition 2.9 A signature scheme Π is a triple of algorithms (/, Sign, Vrfy) such that, for some
polynomial p():
• The key generation algorithm / is a ppt algorithm that takes as input a security parameter k
(in unary) and returns veriﬁcation key V K and signing key SK.
• The signing algorithm Sign is a ppt algorithm that takes as input 1
k
, a key SK, and a message
m ∈ ¦0, 1¦
≤p(k)
and returns a signature s (we denote this by s ← Sign
SK
(m)).
22
• The veriﬁcation algorithm Vrfy is a deterministic algorithm that takes as input 1
k
, a key V K,
a message m ∈ ¦0, 1¦
≤p(k)
, and a signature s and returns a single bit (we denote this by
b = Vrfy
V K
(m, s)).
We require that for all k, all (V K, SK) output by /(1
k
), all m ∈ ¦0, 1¦
≤p(k)
, and all s output by
Sign
SK
(m), we have Vrfy
V K
(m, s) = 1.
Deﬁnition 2.10 A signature scheme Π = (/, Sign, Vrfy) is secure under adaptive chosen message
attack if for all ppt forging algorithms T, the following probability is negligible (in k):
Pr[(V K, SK) ← /(1
k
); (m, s) ← T
Sign
SK
(·)
(1
k
, V K) : Vrfy
V K
(m, s) = 1],
where we require that s not be a signature previously output by Sign
SK
() on input m.
As with macs, the above deﬁnition is standard, so that the term “secure signature scheme” in
this work refers to security in the sense of the deﬁnition above. Secure signature schemes may be
constructed from any one-way function [98, 108].
A weaker notion is that of a one-time signature scheme [87, 94], in which the adversary is
allowed to request only one signature from the Sign oracle before attempting a forgery. Although
secure signature schemes and one-time signature schemes may both be constructed from one-way
functions, known constructions of one-time signature schemes are more eﬃcient [53, 112].
23
Chapter 3
Password-Authenticated Key
Exchange
3.1 Introduction
Protocols for mutual authentication of two parties and generation of a cryptographically-strong
shared key between them (authenticated key exchange) underly most interactions taking place on
the Internet. Indeed, it would be near-impossible to achieve any level of security over an unau-
thenticated network without mutual authentication and key-exchange protocols. The former are
necessary because you always need to know “with whom you are communicating”; the latter are
required because cryptographic techniques (such as encryption, message authentication, etc.) are
useless without a shared cryptographically-strong key which must be periodically refreshed (e.g., for
each new session). Furthermore, high-level protocols are frequently developed and analyzed using
the assumption of “authenticated channels” (see [4] for discussion); yet, this assumption cannot
be realized without a secure mechanism for implementing such channels using previously-shared
information.
We focus here on password-only protocols in which the information previously-shared between
two parties consists only of a short, easily-memorized password.
1
We additionally assume some
public information known to all participants including the adversary attacking the protocol; this
public information may be established by some trusted party or may be generated in some alternate
secure way (e.g., ﬂipping coins publicly). In the password-only setting, it is important to design
protocols which explicitly prevent oﬀ-line dictionary attacks in which an adversary enumerates all
possible passwords, one-by-one, in an attempt to determine the correct password based on previously-
recorded transcripts. Consideration of such attacks is crucial if security is to be guaranteed even
when users of the protocol choose passwords “poorly” (say, from a dictionary of English words).
1
See Chapter 2 for the distinction between “password” and “key”.
24
One might argue that users should be forced to choose high-entropy passwords which cannot be
easily guessed; indeed, the security of an authentication protocol can only improve as the entropy of
the shared secret increases. This recommendation misses the point. Although good password selec-
tion should be encouraged, it remains true that, in practice, user-selected passwords are weak [118].
This is to be expected since high-entropy passwords are diﬃcult (if not impossible) to remember. It
is preferable to recognize this and design protocols which remain secure despite this limitation.
Previous work on password-based protocols [76, 22] (where, in addition to a shared password, a
public-key infrastructure (PKI) is assumed), are a step in the right direction since they recognize
the weakness of passwords selected in practice. However, password-only protocols (even when public
information is assumed) have many practical advantages over password-based protocols. For one,
the password-only model eliminates the need for secure implementation of a PKI, thereby avoiding
the need to deal with issues like user registration, key management, or key revocation (although
these concerns are somewhat mitigated when only servers need certiﬁed public keys). Furthermore,
avoiding the use of a PKI means that an on-line, trusted certiﬁcation authority (CA) is not needed
throughout the lifetime of the protocol; note that the need to access a CA in the public-key setting is
often a performance bottleneck as the number of users becomes large. In the password-only model,
once public information is established new users may join the network at any time and do not
need to inform anyone else of their presence. Finally, in the password-only model no participants
need to know any “secret key” associated with the public parameters. This eliminates the risk that
compromise of a participant will compromise the security of the entire system.
3.1.1 Previous Work
The importance of key exchange as a cryptographic primitive has been recognized since the inﬂu-
ential paper of Diﬃe and Hellman [42]. Soon after, the importance of authenticated key exchange
(and mutual authentication) became apparent. Many protocols for these tasks were proposed (see
[23] for an exhaustive bibliography), followed by increased realization that precise deﬁnitions and
formalizations were necessary. The ﬁrst formal treatments [18, 43, 13, 15, 86, 4, 114] were in a model
in which participants had established cryptographically-strong information in advance of protocol
execution: either a shared key [18, 13, 15, 4, 114] which is used for authentication of messages, or
a public key [4, 114] which is used for encryption or digital signatures. Under these strong setup
assumptions, secure protocols for the two-party [18, 13, 4, 114] and two-party assisted [15] case were
designed and proven secure.
The setting arising most often in practice — in which human users generate and share only weak
passwords — has only recently received formal treatment. It is important to note that the known
25
protocols which guarantee security when users share keys are demonstrably insecure when users
share passwords. For example, a challenge-response protocol in which the client sends a nonce r and
the server replies with x = f
K
(r) (where ¦f
s
¦
s∈{0,1}
k is a family of pseudorandom functions and K
is a shared key) prevents a passive eavesdropper from determining K only when the entropy of K
is suﬃciently large. When K has low entropy, an eavesdropper who monitors a single conversation
(r, x) can determine (with high probability) the value of K, oﬀ-line, by trying all possibilities until
a value K
′
is found such that f
K
′ (r) = x. This example clearly indicates the need for new protocols
in the password-only setting.
In the public-key setting (where, as mentioned above, in addition to sharing a password the client
requires the public key of the server), Lomas et. al [88] were the ﬁrst to present password-based
authentication protocols resistant to oﬀ-line dictionary attacks; these protocols were subsequently
improved [72]. However, formal deﬁnitions and proofs of security are not given. Formal deﬁnitions
and provably-secure protocols for the public-key setting were given by Halevi and Krawczyk [76],
and extensions of these deﬁnitions and protocols to the multi-user setting have also appeared [22].
A protocol for password-only (i.e., where no PKI is assumed) authentication and key exchange
was ﬁrst introduced by Bellovin and Merritt [17], and many additional protocols have subsequently
been proposed [73, 115, 79, 80, 89, 117]. These protocols have only informal arguments for their
security; in fact, some of these protocols were later broken [102] indicating the need for proofs
of security in a well-deﬁned model. Formal models of security for the password-only setting were
given independently by Bellare, Pointcheval, and Rogaway [11] (building on [13, 15, 89]) and Boyko,
MacKenzie, Patel, and Swaminathan [92, 24, 23] (building on [4, 114]); these works also give protocols
for password-only key exchange which are provably-secure in the ideal cipher and random oracle
models, respectively. A diﬀerent model of security for the password-only setting was introduced by
Goldreich and Lindell [65]; they also present a provably-secure protocol under standard assumptions
(i.e., without random oracles or ideal ciphers). Subsequent to the work described in this chapter,
other protocols with provable security in the random oracle model have been demonstrated [90, 91].
3.1.2 Our Contribution
Proofs of security in idealized models (random oracle/ideal cipher) do not necessarily translate to
real-world security [29]. In fact, protocols are known which may be proven secure in an idealized
model yet are demonstrably insecure when given any concrete implementation in the standard model
[29]. This illustrates the importance of proofs of security in the standard model, using well-studied
cryptographic assumptions.
The existence of a secure protocol for password-only key exchange in the standard model [65]
26
is remarkable since it was not a priori clear whether a solution was achievable. In contrast to the
present work, the protocol of Goldreich and Lindell [65] does not require public parameters. On the
other hand (unlike the protocol presented here) their solution does not allow concurrent executions
of the protocol between parties using the same password. Most importantly, their protocol is not at
all eﬃcient. The proposed scheme requires techniques from generic multi-party computation (mak-
ing it computationally ineﬃcient) and concurrent zero-knowledge (making the round-complexity
prohibitive); thus, their protocol may be viewed as a plausibility result that does not settle the
important question of whether a practical solution is possible. We note that eﬃciency is especially
important in the password-only setting since security concerns are motivated by practical consider-
ations (i.e., human users’ inability to remember long keys).
Here, we present a protocol which is provably secure in the standard model under the decisional
Diﬃe-Hellman assumption, a well-studied cryptographic assumption [42] used in constructing pre-
vious password-only schemes [11, 24]. The construction is secure under both the notion of “basic
security” and the stronger notion of “forward security” (see Section 3.2.1 for deﬁnitions). The pro-
tocol is remarkably eﬃcient even when compared to the original key-exchange protocol of Diﬃe
and Hellman [42] which provides no authentication at all. Only three rounds of communication are
needed, and the protocol requires computation only (roughly) 4 times greater than the aforemen-
tioned schemes [42, 11, 24].
Although our solution relies on public-key techniques (in fact, this is necessary [76]), our protocol
does not use the public-key model. In particular, we do not require any participant to have a public
key but instead rely on one set of common parameters shared by everyone in the system. From a
practical point of view, the requirement of public parameters is not a severe limitation. Previous
password-only protocols [11, 24] require public parameters and the existence of such parameters
seems to be implicitly assumed in most previous work.
2
Furthermore, the public parameters can be
hard-coded into any implementation of the protocol. We note, however, that it would be preferable
to rely on public parameters into which no “secret information” can be embedded (e.g., a single
generator g). This makes the problem of generating the public information much easier when no
trusted parties are assumed.
3.2 Deﬁnitions and Preliminaries
We begin with an informal description of the adversarial model, followed by a more formal treatment
in Section 3.2.1. Two parties within a larger network who share a weak (low-entropy) password wish
2
For example, Diﬃe and Hellman [42] implicitly assume that both parties know a group G and generator g to use
in the protocol. Although this can be avoided (these may be included in the ﬁrst message), the public nature of these
parameters has generally been assumed in subsequent work.
27
to authenticate each other and generate a strong session key to secure their future communication.
An adversary controls all communication in the network. The adversary may view, tamper with,
deliver out-of-order, or refuse to deliver messages sent by the honest parties. The adversary may also
initiate concurrent (arbitrarily-interleaved) executions of the protocol between the honest parties;
during these executions, the adversary may attempt to impersonate one (or both) of the parties or
may simply eavesdrop on an honest execution of the protocol. Finally, the adversary may corrupt
the honest parties (in a way we describe below) to expose previous session keys or even the long-term
shared password. The adversary succeeds (informally) if he can distinguish an actual session key
generated by an honest party from a randomly-chosen session key.
A notion of security in this setting must be carefully deﬁned. Indeed, since passwords are chosen
from a small space, an adversary can always try each possibility one at a time in an impersonation
(on-line) attack. We say a password-only protocol is secure (informally) if on-line guessing is the
best an adversary can do. On-line attacks are the hardest to mount, and they are also the easiest to
detect. Furthermore, on-line attacks may be limited by, for example, shutting down a user’s account
after three failed authentication attempts. It is therefore very realistic to assume that the number
of on-line attacks an adversary is allowed is severely limited, while other attacks (eavesdropping,
oﬀ-line password guessing) are not.
3.2.1 The Model
Our model is essentially identical to that proposed by Bellare, Pointcheval, and Rogaway [11] with
a few small diﬀerences. A formal description of the model follows.
Participants, passwords, and initialization. We have a ﬁxed set of protocol participants (also
called principals or users) each of which is either a client C ∈ Client or a server S ∈ Server, where
Client and Server are disjoint. We let User
def
= Client ∪ Server. Each U ∈ User may be viewed as a
string (of length polynomial in the security parameter) identifying that user.
Each C ∈ Client has a password pw
C
. Each S ∈ Server has a vector PW
S
= 'pw
S,C
`
C∈Client
which contains the passwords of each of the clients (we assume that all clients share passwords with
all servers). Recall that pw
C
is what client C remembers for future authentication; therefore, it is
assumed to be chosen from a relatively small space of possible passwords.
Before the protocol is run, an initialization phase occurs during which public parameters are
established and passwords pw
C
are chosen for each client. We assume that passwords for each client
are chosen independently and uniformly
3
at random from the set ¦1, . . . , N¦, where N is a constant
which is ﬁxed independently of the security parameter. The correct passwords are stored at each
3
Our analysis extends easily to handle arbitrary distributions, including users with inter-dependent passwords.
28
server so that pw
S,C
= pw
C
for all C ∈ Client and S ∈ Server.
It is possible for additional information to be generated during this initialization phase. For
example, in the public-key model [76, 22] public/secret key pairs are generated for each server, with
the secret key given as private input to the appropriate server and the public key provided as public
input for all participants. Here, we use the weaker requirement of a set of parameters provided as
public input to all participants. See Chapter 1 for further discussion of these diﬀerent models.
Execution of the protocol. In the real world, a protocol determines how principals behave in
response to signals (input) from their environment. In the model, these signals are sent by the
adversary. Each principal is able to execute the protocol multiple times with diﬀerent partners;
this is modeled by allowing each principal to have an unlimited number of instances with which to
execute the protocol [15]. We denote instance i of user U as Π
i
U
. A given instance may be used only
once. Each instance Π
i
U
has associated with it various variables:
• state
i
U
denotes the state of the instance including any information necessary for execution of
the protocol. During the initialization stage, this variable is set to null for all instances.
• term
i
U
is a boolean variable denoting whether a given instance has terminated (i.e., is done
sending and receiving messages); during the initialization phase, this variable is set to false
for all instances.
• acc
i
U
is a boolean variable denoting whether a given instance has accepted, where acceptance
is deﬁned by the protocol speciﬁcation. When an instance accepts, the values of sid
i
U
, pid
i
U
,
and sk
i
U
(see below) are non-null. As an example, acceptance might indicate that instance Π
i
U
is convinced of the identity of the user with whom it was interacting. During the initialization
phase, this variable is set to false for all instances.
• used
i
U
is a boolean variable denoting whether a given instance has begun executing the protocol;
this variable is used to ensure that a given instance is used only once. During the initialization
phase, this variable is set to false for all instances.
• sid
i
U
, pid
i
U
, and sk
i
U
are variables containing the session id, partner id, and session key for an
instance, respectively. Computation of the session key is the goal of the protocol, and this
key will be used to secure future communication between the interacting parties. The precise
function of sid
i
U
and pid
i
U
will be explained in more detail below. During the initialization
phase, these variables are set to null for all instances.
The adversary is assumed to have complete control over all communication in the network. The
adversary’s interaction with the principals (more speciﬁcally, with the various instances) is modeled
29
via access to oracles which we describe in detail below. Local state (in particular, values for the
variables described above) is maintained for each instance with which the adversary interacts; this
state is not directly visible to the adversary. The state for an instance may be updated during an
oracle call, and the oracle’s output may depend upon this state. The oracle types are:
• Send(U, i, M) — This sends message M to instance Π
i
U
. The oracle runs this instance according
to the protocol speciﬁcation, maintaining state as appropriate. The output of Π
i
U
(i.e., the
message sent by the instance) is given to the adversary, and in addition the adversary receives
the updated values of sid
i
U
, pid
i
U
, acc
i
U
, and term
i
U
.
• Execute(C, i, S, j) — If Π
i
C
and Π
j
S
have not yet been used (where C ∈ Client and S ∈ Server),
this oracle executes the protocol between these instances and outputs the transcript of this
execution. This oracle call represents occasions when the adversary passively eavesdrops on a
protocol execution. In addition to the transcript, the adversary receives the values of sid, pid,
acc, and term, for both instances, at each step of protocol execution.
• Reveal(U, i) — This outputs the current value of session key sk
i
U
. This oracle call models
possible leakage of session keys due to, for example, improper erasure of session keys after use,
compromise of a host computer, or cryptanalysis.
• Corrupt(C, (pw, S)) — This oracle call outputs pw
C
(where C ∈ Client) and, if pw =⊥, sets
pw
S,C
= pw. This represents possible password exposures and also gives the adversary the
ability to install bogus passwords of his choice on the various servers.
• Test(U, i) — This query is allowed only once, at any time during the adversary’s execution. A
random bit b is generated; if b = 1 the adversary is given sk
i
U
, and if b = 0 the adversary is
given a random session key. This oracle call does not correspond to any real-world event, but
will allow us to deﬁne a notion of security.
Speciﬁcation of the Corrupt oracle, above, corresponds to the “weak-corruption” case [11]; in the
“strong-corruption” case the adversary also obtains the state information for all instances associated
with C.
Correctness. Any key-exchange protocol must satisfy the following notion of correctness: Let
C ∈ Client and S ∈ Server. If two instances Π
i
C
and Π
j
S
satisfy sid
i
C
= sid
j
S
and acc
i
C
= acc
j
S
= true,
and furthermore it was the case that pw
C
= pw
S,C
throughout the time these instances were active,
then it must be the case that sk
i
C
= sk
j
S
.
30
Partnering. We say that two instances Π
i
U
and Π
j
U
′ are partnered if: (1) U ∈ Client and U
′
∈ Server,
or U ∈ Server and U
′
∈ Client; (2) sid
i
U
= sid
j
U
′ = null; (3) pid
i
U
= U
′
and pid
j
U
′ = U; and (4)
sk
i
U
= sk
j
U
′ . The notion of partnering will be fundamental in deﬁning the notion of security.
Advantage of the adversary. Informally, the adversary succeeds if it can guess the bit b used by
the Test oracle. Before formally deﬁning the adversary’s success, we deﬁne a notion of freshness. The
adversary can succeed only if the Test query was made for an instance which is fresh at the end of the
adversary’s execution. This is necessary for any reasonable deﬁnition of security; if the adversary’s
behavior were unrestricted the adversary could always succeed by, for example, submitting a Test
query for an instance for which it had already submitted a Reveal query.
Two notions of freshness may be deﬁned, one for the “basic” case and one for the “forward-secure”
case. An instance Π
i
U
is fresh (in the basic case) unless one of the following is true:
• At some point, the adversary queried Reveal(U, i).
• At some point, the adversary queried Reveal(U
′
, j) where Π
j
U
′ and Π
i
U
are partnered.
• At some point, the adversary queried the Corrupt oracle.
For the case of forward security, an instance Π
i
U
is fs-fresh unless one of the following is true:
• At some point, the adversary queried Reveal(U, i).
• At some point, the adversary queried Reveal(U
′
, j) where Π
j
U
′ and Π
i
U
are partnered.
• The adversary makes the Test query after a Corrupt query and at some point the adversary
queried Send(U, i, M) for some M.
In the basic case, adversary / succeeds if it makes a single query Test(U, i) to the Test oracle,
where acc
i
U
and term
i
U
are true
4
at the time of this query and Π
i
U
is fresh, outputs a single bit b
′
,
and b
′
= b (where b is the bit chosen by the Test oracle). We denote this event by Succ. For the
forward-secure case, / succeeds if it makes a single query Test(U, i) to the Test oracle, where acc
i
U
was true at the time of this query and Π
i
U
is fs-fresh, outputs a single bit b
′
, and b
′
= b. We denote
this event by fsSucc. The advantage of adversary / in attacking protocol P in the basic sense is
deﬁned by:
Adv
A,P
(k)
def
= 2 Pr[Succ] −1.
A similar deﬁnition may be given for fsAdv
A,P
(k) in the forward secure case. Note that probabilities
of success are functions of the security parameter k, and are taken over the random coins used by the
4
For the protocol presented here, acc
i
U
= true automatically implies that term
i
U
= true.
31
adversary, random coins used during the initialization phase, and random coins used by the various
oracles during the actual experiment.
We have not yet deﬁned what we mean by a secure protocol. Note that a ppt adversary can
always succeed by trying all passwords one-by-one in an on-line impersonation attack (recall that
the number of possible passwords is constant). Informally, we say a protocol is secure if this is the
best an adversary can do. More formally, an instance Π
i
U
represents an on-line attack
5
if both the
following are true:
• At some point, the adversary queried Send(U, i, M) for some M.
• At some point, the adversary queried Reveal(U, i) or Test(U, i).
In particular, instances with which the adversary interacts via Execute calls are not counted as on-
line attacks. The number of on-line attacks represents a bound on the number of passwords the
adversary could have tested in an on-line fashion. This motivates the following deﬁnition (with a
similar deﬁnition for the case of forward security):
Deﬁnition 3.1 Protocol P is a secure password-only key-exchange protocol (in the basic sense) if,
for all N and for all ppt adversaries / making at most Q(k) on-line attacks, there exists a negligible
function ε() such that:
Adv
A,P
(k) ≤ Q(k)/N +ε(k).
In particular, this indicates that the adversary can (essentially) do no better than guess a single
password during each on-line attempt. Calls to the Execute oracle, which are not included in the
count Q(k), are of no help to the adversary in breaking the security of the protocol; this means that
passive eavesdropping and oﬀ-line dictionary attacks are of (essentially) no use.
Some previous deﬁnitions of security for password-only key-exchange protocols [11, 65] consider
protocols secure as long as the adversary can do no better than guess a constant number of passwords
in each on-line attempt. We believe the strengthening given by Deﬁnition 3.1 (in which the adversary
can guess only a single password per on-line attempt) is an important one. The space of possible
passwords is small to begin with, so any degradation in security should be avoided if possible. This
is not to say that protocols which do not meet this deﬁnition of security should never be used;
however, before using such a protocol, one should be aware of the constant implicit in the proof of
security.
An examination of the security proofs for some protocols [11, 24, 92] shows that these protocols
achieve the stronger level of security given by Deﬁnition 3.1. However, security proofs for other
5
The deﬁnition of an on-line attack given here is valid only for key-exchange protocols without explicit authenti-
cation.
32
protocols [65, 91] are inconclusive, and leave open the possibility that more than one password can
be guessed by the adversary in a single on-line attack. In at least one case [117], an explicit attack
is known which allows an adversary to guess two passwords per on-line attack.
3.2.2 Protocol Components
We discuss some cryptographic tools used to construct our protocol.
Basic primitives. The security of our protocol relies on the DDH assumption [42], introduced
in Section 2.5.2. We also use universal one-way hash families [98] and one-time digital signature
schemes, as discussed in Section 2.5.3. It should be noted, however, that these may both be con-
structed from any one-way function [108]; in particular, the DDH assumption (which implies that
group exponentiation is one-way) implies the existence of universal one-way hash families and one-
time signature schemes.
Non-malleable commitment. Our protocol requires a particular non-malleable commitment
scheme that we construct based on the Cramer-Shoup [36] cryptosystem, whose security against
chosen-ciphertext attacks (cf. Deﬁnition 2.5) relies on the DDH assumption. It is interesting that
chosen-ciphertext-secure public-key encryption has been used previously to construct authentication
and key exchange protocols [4, 76, 22, 114]. We stress that our protocol diﬀers from these works in
that we use the scheme for commitment only and not for encryption. No party publishes their own
public key and no one need hold any secret key; in fact, “decryption” is never performed during
execution of the protocol.
As we point out in Chapter 4, any CCA2 public-key encryption scheme immediately yields a
non-malleable commitment scheme when public parameters are available to all parties. We may
therefore describe our modiﬁed scheme as a public-key encryption scheme since this form is simplest
for the proof of security; we emphasize that it is used only as a commitment scheme in the actual
key-exchange protocol. Key generation proceeds by running ((1
k
) to yield primes p, q deﬁning
group G in which the DDH assumption is assumed to hold. Random values g
1
, g
2
∈ G are selected,
along with random z, x
1
, x
2
, y
1
, y
2
∈ Z
q
. Additionally, sets Client and Server of polynomial size (in
k) are ﬁxed; these sets contain strings which will be necessary for the key-exchange protocol but
whose exact structure is unimportant here. Finally, a random hash function H is chosen from a
family of universal hash functions H
k
. The public key is pk = (G, g
1
, g
2
, h = g
z
1
, c = g
x1
1
g
x2
2
, d =
g
y1
1
g
y2
2
, H, Client, Server).
Ciphertexts are of the form 'A[B[C[D[E[F`, where C, D, E, F ∈ G and the purpose of A, B will
be described below. The actions of the decryption oracle T
sk
() are as in the original scheme [36]:
33
ﬁrst, α = H(A, B, C, D, E) is computed, and the following condition is checked:
C
x1+y1α
D
x2+y2α
?
= F.
If this fails, the output is ⊥. (We also output ⊥ if any of C, D, E, F are not in G; note that this
may be eﬃciently veriﬁed.) Otherwise, output the plaintext E/C
z
.
The central modiﬁcation we introduce is in the deﬁnition of encryption, and more precisely in the
deﬁnition of the encryption oracle to which the adversary will have access. Besides sending messages
m
0
, m
1
∈ G to the encryption oracle, the adversary also includes a bit t specifying an encryption-type
which is either client-encryption or server-encryption. Depending on the encryption-type selected,
the adversary also submits some additional information which is used in the encryption. For a client-
encryption, the adversary includes a value Client ∈ Client; for a server-encryption, the adversary
includes values Server ∈ Server and α ∈ Z
q
. The encryption oracle chooses a random bit b and
encrypts message m
b
according to the requested encryption-type. The encryption oracle outputs
the resulting ciphertext along with some additional information. Formally, the encryption oracle is
deﬁned as follows:
O
1
k
,pk,b
(m
0
, m
1
, t, input)
if t = 0 and input ∈ Client then
Client-encryption(1
k
, pk, m
b
, input)
if t = 1 and input ∈ Server Z
q
then
Server-encryption(1
k
, pk, m
b
, input)
where Client-encryption and Server-encryption are deﬁned by:
Client-encryption(1
k
, pk, m, Client)
(VK, SK) ← /(1
k
)
A := Client; B := VK
r ←Z
q
C := g
r
1
; D := g
r
2
; E := h
r
m
α := H(A[B[C[D[E)
F := (cd
α
)
r
return('A[B[C[D[E[F`, SK)
Server-encryption(1
k
, pk, m, (Server, α))
x, y, z, w, r ←Z
q
A := Server; B := g
x
1
g
y
2
h
z
(cd
α
)
w
C := g
r
1
; D := g
r
2
; E = h
r
m
β := H(A[B[C[D[E)
F := (cd
β
)
r
return('A[B[C[D[E[F`, (x, y, z, w))
(here, / is a key-generation algorithm for a secure one-time signature scheme).
Let Gen denote the key-generation algorithm for this scheme. For any adversary / = (/
1
, /
2
),
deﬁne the adversary’s advantage Adv
nm
A
(k) in guessing the bit b used by the encryption oracle as
Adv
nm
A
(k)
def
=

,
where /
2
may not submit C to the decryption oracle.
Lemma 3.1 Under the DDH assumption, Adv
nm
A
(k) is negligible for any ppt adversary /.
34
Sketch of Proof The proof of security exactly follows that given by Cramer and Shoup [36], and
it can be veriﬁed easily that the additional information info given to the adversary does not improve
her advantage. One point requiring careful consideration is the adversary’s probability of ﬁnding a
collision for the hash function H included in the public key. If H is chosen from a collision-resistant
hash family (a stronger assumption than being universal one-way), there is nothing left to prove. If
H is universal one-way, the value t and an appropriate value of Client or Server can be guessed by
the encryption oracle in advance of the adversary’s query; an appropriate value for B, based on the
guess for t, can also be generated in advance (in one case by running the key-generation algorithm
/(1
k
) for the one-time signature scheme and in the other case by choosing random x ∈ Z
q
and
setting B = g
x
1
). In particular, these values may all be determined before the encryption oracle is
given the function H. Although the encryption oracle cannot guess the value α in advance, this
will not present a problem since the encryption oracle can provide a random representation of B
with respect to (g
1
, g
2
, h, (cd
α
)) for any value α output by the adversary as long as the encryption
oracle knows the discrete logarithms of g
2
, h, c, d with respect to g
1
. This can be ensured during key
generation.
The encryption oracle ﬁnds a collision for H if (1) the oracle correctly guesses the values t and
Client /Server (depending on t); and (2) the adversary ﬁnds a collision for H. The probability of
guessing the adversary’s choices correctly is at least
1
2·max{|Client|,|Server|}
; since [Client[ and [Server[ are
polynomial in k, this probability is an inverse polynomial in k. Thus, if the adversary’s advantage
in ﬁnding a collision in the experiment is non-negligible, this implies a non-negligible advantage in
ﬁnding a collision in H.
Using a standard hybrid argument, Lemma 3.1 implies that Adv
nm
A
(k) is negligible for any ppt
adversary / that queries the encryption oracle polynomially-many times (with arbitrary values of
m
0
, m
1
, t, input), as long as the adversary may not submit any of the returned ciphertexts to the
decryption oracle.
Since we use the scheme for commitment within our protocol, we refer to Client-commitment
and Server-commitment of a message m. The mechanism for this is exactly as sketched above (and
decommitment is not needed in the present context).
3.3 Protocol Details
A high-level description of the protocol is given in Figure 3.1, and a more detailed description follows
here. A formal speciﬁcation of the protocol appears in Section 3.4.
During the initialization phase, public information is established. Given a security parameter k,
primes p, q are chosen such that [q[ = k and p = 2q + 1 using algorithm (; these values deﬁne a
35
Public: G, g
1
, g
2
, h, c, d ∈ G; H : ¦0, 1¦
∗
→Z
q
Client Server
(VK, SK) ← /(1
k
)
r
1
←Z
q
A := g
r1
1
; B := g
r1
2
C := h
r1
pw
C
α := H(Client [VK[A[B[C)
D := (cd
α
)
r1
Client [ VK [ A [ B [ C [ D
-
x
2
, y
2
, z
2
, w
2
, r
2
←Z
q
α
′
:= H(Client [VK[A[B[C)
E := g
x2
1
g
y2
2
h
z2
(cd
α
′
)
w2
F := g
r2
1
; G := g
r2
2
I := h
r2
pw
C
β := H(Server [E[F[G[I)
J := (cd
β
)
r2
Server [ E [ F [ G [ I [ J

(if the random tuple satisﬁes log
g
s = log
h
t, the distribution on the view of A in the above experiment
is equivalent to the distribution on the view of A in experiment P
′
0
; for a random tuple, this occurs
with probability 1/q). The claim follows from the observation that this advantage is negligible under
the DDH assumption and from the fact that 1/q is negligible.
In experiment P
2
, the simulator interacts with the adversary as in P
1
except that during queries
Execute(Client, i, Server, j) the session key sk
i
Client
is chosen uniformly at random from G; session
key sk
j
Server
is set equal to sk
i
Client
.
Claim 3.3 [Adv
A,P1
(k) −Adv
A,P2
(k)[ ≤ ε(k) for some negligible function ε().
The claim follows from the negligible statistical diﬀerence between the distributions on the adver-
sary’s view in the two experiments. In P
1
, elements C and I are chosen at random and the session
keys are computed as in (3.1). Assuming that h is a generator, we may write C = h
r
′
1
pw
Client
and
I = h
r
′
2
pw
Client
for some r
′
1
, r
′
2
∈ Z
q
. With all but negligible probability 1/q
2
, we have either r
′
1
= r
1
or r
′
2
= r
2
. Assume the former. For any µ, ν ∈ G and ﬁxing the random choices for the remainder
of experiment P
1
, the probability over choice of x
2
, y
2
, z
2
, w
2
that E = µ and sk
i
Client
= ν is exactly
the probability that
log
g1
µ = x
2
+y
2
log
g1
g
2
+z
2
log
g1
h +w
2
log
g1
(cd
α
) (3.2)
and
log
g1
ν −log
g1
(F
x1
G
y1
(I/pw
Client
)
z1
J
w1
) =
x
2
r
1
+y
2
r
1
log
g1
g
2
+z
2
r
′
1
log
g1
h +w
2
r
1
log
g1
(cd
α
) (3.3)
where we assume g
1
is a generator (if this is not the case the experiment is aborted). Viewing (3.2)
and (3.3) as equations in x
2
, y
2
, z
2
, w
2
we see that they are linearly independent and not identically
zero whenever r
′
1
= r
1
(here, we use the fact that h is a generator and therefore log
g1
h = 0), the
desired probability is 1/q
2
. In other words, when r
′
1
= r
1
the value of sk
i
Client
is independent of the
value of E and hence independent of the remainder of experiment P
1
. A similar argument shows
that when r
′
2
= r
2
, the value of sk
i
Client
is independent of K and hence independent of the rest of
experiment P
1
.
46
Initialize(1
k
) —
(p, g) ← ((1
k
)
g
1
, g
2
←G
χ
1
, χ
2
, ξ
1
, ξ
2
, κ ←Z
q
h := g
κ
1
; c := g
χ1
1
g
χ2
2
; d := g
ξ1
1
g
ξ2
2
H ← UOWH(1
k
)
(Client, Server) ← UserGen(1
k
)
for each C ∈ Client
pw
′
C
← ¦1, . . . , N¦
pw
C
:= g
pw
′
C
1
for each S ∈ Server
pw
S,C
:= pw
C
return Client, Server, G, g
1
, g
2
, h, c, d, H
Figure 3.6: Modiﬁed initialization procedure.
Thus, the adversary’s view in P
1
is distributed identically to the adversary’s view in P
2
assuming
that, for all Execute queries, either r
′
1
= r
1
or r
′
2
= r
2
. For a particular Execute query, this condition
holds except with negligible probability 1/q
2
. Since the adversary is permitted to query the Execute
oracle only polynomially-many times, the claim follows.
Before continuing, we introduce some notation. For a query Send
1
(Server, j, msg-in), where
msg-in = 'Client[VK[A[B[C[D`, we say that msg-in is previously-used if it was ever previously
output by a Send
0
oracle. Similarly, when the adversary queries Send
2
(Client, i, msg-in), where
msg-in = 'Server[E[F[G[I[J`, we say that msg-in is previously-used if it was ever previously output
by a Send
1
oracle. A msg-in for either a Send
1
or Send
2
oracle query which is not previously-used
is called new.
In experiment P
3
, the simulator runs the modiﬁed initialization procedure shown in Figure 3.6,
where the values χ
1
, χ
2
, ξ
1
, ξ
2
, κ are stored for future use. Furthermore, queries to the Send
2
oracle
are handled diﬀerently. Upon receiving query Send
2
(Client, i, 'Server[E[F[G[I[J`), the simulator
examines msg-in. If msg-in is previously-used, the query is answered as in experiment P
2
. If msg-in
is new, the simulator checks whether F
χ1+βξ1
G
χ2+βξ2
?
= J and I/pw
Client
?
= F
κ
. If not, the query
is said to appear invalid and is answered as in experiment P
2
. Otherwise, msg-in is said to appear
valid; the query is answered as in experiment P
2
except that if sk
i
Client
is to be assigned a value, it
is assigned the special value ∇.
Queries to the Send
3
oracle are also handled diﬀerently. Upon query Send
3
(Server, j, msg-in),
the simulator examines ﬁrst-msg-in = 'Client[VK[A[B[C[D`, the message sent to the Send
1
oracle
for the same instance; note that if ﬁrst-msg-in is not deﬁned, the query to Send
3
simply returns
⊥ as in experiment P
2
. If ﬁrst-msg-in is previously-used, the query is answered as in experiment
47
P
2
. If ﬁrst-msg-in is new, the simulator computes α = H(Client[VK[A[B[C) and checks whether
A
χ1+αξ1
B
χ2+αξ2
?
= D and C/pw
Client
?
= A
κ
. If not, ﬁrst-msg-in is said to appear invalid and the
query is answered as in experiment P
2
. Otherwise, ﬁrst-msg-in is said to appear valid and the query
is answered as in experiment P
2
except that if sk
j
Server
is to be assigned a value, it is assigned the
special value ∇.
Finally, the deﬁnition of the adversary’s success is changed. If the adversary ever queries
Reveal(U, i) or Test(U, i) where sk
i
U
= ∇, the simulator halts execution and the adversary immedi-
ately succeeds. Otherwise, the adversary’s success is determined as in experiment P
2
.
Claim 3.4 Adv
A,P2
(k) ≤ Adv
A,P3
(k).
The probability that g
1
and g
2
are both generators is the same in experiments P
2
and P
3
. Condi-
tioned on the event that g
1
and g
2
are generators (if not, the experiment is aborted), the distribu-
tions on the adversary’s views in experiments P
2
and P
3
are identical until the adversary queries
Reveal(U, i) or Test(U, i) where sk
i
U
= ∇; if such a query is never made, the distributions on the
views are identical. The claim follows immediately since there are more ways for the adversary to
succeed in experiment P
3
.
In experiment P
4
, queries to the Send
3
oracle are handled diﬀerently. First, whenever the sim-
ulator responds to a Send
2
query, the simulator stores the values (K, β, x, y, z, w), where K =
g
x
1
g
y
2
h
z
(cd
β
)
w
. Upon receiving query Send
3
(Server, j, 'K[Sig`), the simulator checks the value of
ﬁrst-msg-in = 'Client[VK[A[B[C[D` (if ﬁrst-msg-in is not deﬁned the query to Send
3
simply re-
turns ⊥ as in experiment P
3
). If ﬁrst-msg-in is new, the query is answered as in experiment P
3
. If
ﬁrst-msg-in is previously-used and Vrfy
VK
(β[K, Sig) = 1, the query is answered as in experiment P
3
and the session key is not assigned a value. If ﬁrst-msg-in is previously-used, Vrfy
VK
(β[K, Sig) = 1,
and the experiment is not aborted, the simulator ﬁrst checks whether there exists an i such that
sid
i
Client
= sid
j
Server
(if such an i exists it must be unique since the experiment is aborted if a ver-
iﬁcation key VK repeats during the experiment). If so, sk
j
Server
(if it is assigned a value at all) is
assigned the value sk
i
Client
. Otherwise, let ﬁrst-msg-out = 'Server[E[F[G[I[J`. The simulator must
have stored values x
′
, y
′
, z
′
, w
′
such that K = g
x
′
1
g
y
′
2
h
z
′
(cd
β
)
w
′
(this is true since the experiment is
aborted if Sig is a valid signature on β[K that was not output by the simulator following a Send
2
query). The session key (assuming it is assigned a value at all) is then assigned the value:
sk
j
Server
:= A
x
B
y
(C/pw
Client
)
z
D
w
F
x
′
G
y
′
(I/pw
Client
)
z
′
J
w
′
.
Claim 3.5 Adv
A,P4
(k) = Adv
A,P3
(k).
The distribution on the adversary’s view is identical in experiments P
3
and P
4
. Indeed, when ﬁrst-
msg-in is previously used, Vrfy
VK
(β[K, Sig) = 1, and there exists an i as described above, it is always
48
the case that sk
j
Server
= sk
i
Client
. When ﬁrst-msg-in is previously used, Vrfy
VK
(β[K, Sig) = 1, and
an i such that sid
j
Server
= sid
i
Client
does not exist, then as long as sk
j
Server
is to be assigned a value
it is the case that K
r
= F
x
′
G
y
′
(I/pw
Client
)
z
′
J
w
′
, where x
′
, y
′
, z
′
, w
′
are as above. Therefore, the
session key computed in P
4
matches the session key that would have been computed in P
3
.
In experiment P
5
, queries to the Send
3
oracle are handled diﬀerently. Upon receiving query
Send
3
(Server, j, 'K[Sig`), the simulator checks the value of ﬁrst-msg-in (if ﬁrst-msg-in is not deﬁned
the simulator returns ⊥ as in experiment P
4
). If ﬁrst-msg-in is new and appears invalid and the
session key is to be assigned a value, the session key is assigned a value randomly chosen in G.
Otherwise, the query is answered as in experiment P
4
.
Claim 3.6 Adv
A,P5
(k) = Adv
A,P4
(k).
The claim will follow from the equivalence of the distributions on the adversary’s view in the two
experiments. For a given query Send
3
(Server, j, msg-in) where ﬁrst-msg-in = 'Client[VK[A[B[C[D`
is new and appears invalid, let ﬁrst-msg-out = 'Server[E[F[G[I[J` and α = H(Client[VK[A[B[C).
Since ﬁrst-msg-in appears invalid, it must be the case that either A
χ1+αξ1
B
χ2+αξ2
= D or else
C/pw
Client
= A
κ
(or possibly both). For any µ, ν ∈ G and ﬁxing the randomness used in the rest of
experiment P
4
, the probability over choice of x, y, z, w that E = µ and sk
j
Server
= ν is exactly the
probability that
log
g1
µ = x +y log
g1
g
2
+z log
g1
h +w log
g1
(cd
α
) (3.4)
and
log
g1
ν −r log
g1
K = x log
g1
A+ y log
g1
B +z log
g1
(C/pw
Client
) +w log
g1
D, (3.5)
where we use the fact that g
1
is a generator (if not, the experiment is aborted). If log
g1
A = 0, it can
be veriﬁed immediately that (3.4) and (3.5) are linearly independent and not identically zero (this
last fact follows by noting that log
g1
A = 0 implies log
g1
(C/pw
Client
) = 0). If log
g1
A = 0, it can
be similarly veriﬁed that (3.4) and (3.5) are linearly independent and not identically zero. In either
case, then, the desired probability is 1/q
2
. In other words, the value of sk
j
Server
is independent of
the value of E and hence independent of the remainder of the experiment.
In experiment P
6
, queries to the Send
1
oracle are handled diﬀerently. Now, I is computed as
h
r
g
N+1
1
, where the dictionary of legal passwords is ¦1, . . . , N¦; note that g
N+1
1
represents an invalid
password since N < q −1.
Claim 3.7 Under the DDH assumption, [Adv
A,P5
(k) − Adv
A,P6
(k)[ ≤ ε(k), for some negligible
function ε().
49
The claim follows from the non-malleability of the commitment scheme used (i.e., the Cramer-Shoup
encryption scheme). We show that the simulator can use A as a subroutine in order to distinguish
encryptions of the correct client password(s) from encryptions of g
N+1
1
. The simulator is given a
public key pk = 'G, g
1
, g
2
, h, c, d, H` for an instance of the Cramer-Shoup encryption scheme and
may repeatedly query an encryption oracle O
1
k
,pk,

b
(, , , ) where
¯
b is a randomly-chosen bit. The
simulator may also query a decryption oracle T
sk
() using any ciphertext except those received from
its encryption oracle. The advantage of the simulator is half the absolute value of the diﬀerence
between the probability the simulator outputs 1 when
¯
b = 1 and the probability the simulator
outputs 1 when
¯
b = 0.
The simulator begins by running the following modiﬁed initialization protocol:
Initialize
′
(1
k
, G, g
1
, g
2
, h, c, d, H) —
(Client, Server) ← UserGen(1
k
)
for each C ∈ Client
pw
′
C
← ¦1, . . . , N¦
pw
C
:= g
pw
′
C
1
for each S ∈ Server
pw
S,C
:= pw
C
return Client, Server, G, g
1
, g
2
, h, c, d, H
The simulator responds to Send
0
, Send
2
, Execute, Reveal, and Test oracle queries as in experiments
P
5
, P
6
. The simulator responds to Send
1
and Send
3
queries as shown in Figure 3.7 (the simulator’s
response to Send
3
queries is the same as in experiments P
5
, P
6
, but is included in Figure 3.7 for
convenience). In particular, when a response to a Send
1
query is needed, the simulator queries the
encryption oracle, requesting a server-encryption of either the correct password or the value g
N+1
1
.
To respond to a Send
3
query, the simulator checks whether ﬁrst-msg-in is previously-used or new. If
ﬁrst-msg-in is new, the server determines whether it appears valid or appears invalid by submitting
ﬁrst-msg-in to the decryption oracle. Note that the simulator never need submit to the decryption
oracle a ciphertext that it received from the encryption oracle. The simulator outputs 1 if and only
if A succeeds.
Examination of Figure 3.7 shows that when
¯
b = 0 the distribution on the view of A throughout
the experiment is equivalent to the distribution on the view of A in experiment P
5
. On the other
hand, when
¯
b = 1 the distribution on the view of A throughout the experiment is equivalent to the
distribution on the view of A in experiment P
6
. The simulator’s advantage is therefore
1
2
[Pr
A,P5
[Succ] −Pr
A,P6
[Succ][.
The claim follows from the observation that this advantage is negligible under the DDH assumption
(cf. Lemma 3.1).
50
Send
1
(Server, j, 'Client[VK[A[B[C[D`) —
if Server / ∈ Server or used
j
Server
return ⊥
used
j
Server
:= true
if A, B, C, D / ∈ G or Client / ∈ Client
status
j
Server
:= 'null, null, false, true`; return status
j
Server
α := H(Client[VK[A[B[C)
('Server[E[F[G[I[J`, (x, y, z, w)) ← O
1
k
,pw,

b
(pw
Client
, g
N+1
1
, 1, (Server, α))
β := H(Server[E[F[G[I); msg-out := 'Server[E[F[G[I[J`
status
j
Server
:= 'null, Client, false, false`
state
j
Server
:= 'msg-in, x, y, z, w, β, msg-out, pw
Server,Client
`
return msg-out, status
j
Server
Send
3
(Server, j, 'K[Sig`) —
if Server / ∈ Server or not used
j
Server
or term
j
Server
return ⊥
'ﬁrst-msg-in, x, y, z, w, β, ﬁrst-msg-out, pw` := state
j
Server
'Client[VK[A[B[C[D` := ﬁrst-msg-in
if K / ∈ G or Vrfy
VK
(β[K, Sig) = 1
status
j
Server
:= 'null, null, false, true`; return status
j
Server
sid
j
Server
:= 'ﬁrst-msg-in[ﬁrst-msg-out[msg-in`
acc
j
Server
:= term
j
Server
:= true
if ﬁrst-msg-in is previously-used
if there exists an i such that sid
i
Client
= sid
j
Server
sk
j
Server
:= sk
i
Client
else
retrieve x
′
, y
′
, z
′
, w
′
such that K = g
x
′
1
g
y
′
2
h
z
′
(cd
β
)
w
′
(if such values are not stored, abort the experiment)
sk
j
Server
:= A
x
B
y
(C/pw)
z
D
w
F
x
′
G
y
′
(I/pw)
z
′
J
w
′
else
pw
′
:= T
sk
(ﬁrst-msg-in)
if pw
′
= pw
Client
then sk
j
Server
:= ∇
if pw
′
= pw
Client
then sk
j
Server
←G
return status
j
Server
Figure 3.7: The modiﬁed Send
1
and Send
3
oracles for the proof of Claim 3.7.
51
In experiment P
7
, queries to the Send
2
oracle are handled diﬀerently. Whenever msg-in is new and
appears invalid or is previously-used, the session key (if it is assigned a value at all) is assigned a value
chosen randomly from G. Queries to the Send
3
oracle are also handled diﬀerently. If ﬁrst-msg-in is
previously-used, Vrfy
VK
(β[K, Sig) = 1, and there does not exist an i such that sid
i
Client
= sid
j
Server
,
then the session key is assigned a value chosen randomly from G.
Claim 3.8 Adv
A,P7
(k) = Adv
A,P6
(k).
The claim follows from the equivalence of the distributions on the adversary’s views in the two experi-
ments. First consider a particular query Send
2
(Client, i, msg-in) where msg-in = 'Server[E[F[G[I[J`
is previously-used. Let β = H(Server[E[F[G[I) and r = log
g1
E. Since msg-in is previously-used,
it was output in response to some query to the Send
1
oracle; therefore, regardless of Client we have
F = g
r
1
, F = g
r
2
, J = (cd
β
)
r
, and I = h
r
′
pw
Client
for some r
′
= r. In particular, I/pw
Client
= F
κ
.
A proof similar to that of Claim 3.6 indicates that sk
i
Client
is uniformly distributed independent of
the rest of the experiment.
If the simulator responds to a query Send
2
(Client, i, msg-in) and msg-in = 'Server[E[F[G[I[J`
is new and appears invalid, let 'K[Sig` be the message output by the simulator in responding to
this query. There are two cases to consider. In the ﬁrst case, the values x, y, z, w used by instance
Π
i
Client
are used to compute only K and sk
i
Client
(and, in particular, are never used to compute a
session key sk
j
Server
during a Send
3
query). In this case, an argument exactly as in the proof of
Claim 3.6 shows that the value of sk
i
Client
is independent of the value K and hence independent of
the remainder of the experiment. In the second case, the values x, y, z, w are used at some point
to compute a session key sk
j
Server
when the simulator responds to a Send
3
query. Note that these
values can be used at most once to compute a session key during a Send
3
query, since the session
key is assigned a value only if Vrfy
VK
(β[K, Sig) = 1 and the experiment is aborted if a value β used
by the simulator in responding to a Send
1
query is used twice. We show that in this case the joint
distribution on (K, sk
i
Client
, sk
j
Server
) is uniform, independent of the rest of the experiment.
Let
sid
i
Client
= 'Client[VK[A[B[C[D[Server[E[F[G[I[J[K[Sig`
and
sid
j
Server
= 'Client
′
[VK
′
[A
′
[B
′
[C
′
[D
′
[Server
′
[E
′
[F
′
[G
′
[I
′
[J
′
[K[Sig`,
where the same randomly-chosen values x, y, z, w are used during computation of K, sk
i
Client
, and
sk
j
Server
. Since veriﬁcation keys used by the simulator in responding to Send
0
queries do not repeat
and 'Client
′
[VK
′
[A
′
[B
′
[C
′
[D
′
` is previously-used (if not, x, y, z, w are not used to compute sk
j
Server
),
it must be the case that 'Client[VK[A[B[C[D` = 'Client
′
[VK
′
[A
′
[B
′
[C
′
[D
′
`. Furthermore, we must
52
have 'Server[E[F[G[I` = 'Server
′
[E
′
[F
′
[G
′
[I
′
` (otherwise a collision in H has been found and the
experiment is aborted) and J = J
′
(otherwise sid
j
Server
= sid
i
Client
and x, y, z, w are not used
to compute sk
j
Server
). Denote pw
Client
by pw
C
and let log() denote log
g1
() (recall that g
1
is a
generator). For any µ, ν
1
, ν
2
∈ G and ﬁxing the randomness used in the rest of experiment P
6
,
the probability over choice of x, y, z, w that K = µ, sk
i
Client
= ν
1
, and sk
j
Server
= ν
2
is exactly the
probability that
log µ = x +y log g
2
+z log h +w log(cd
β
)
log ν
1
−r log E = x log F +y log G+z log(I/pw
C
) +w log J
and
log ν
2
−log(A
x
′
B
y
′
(C/pw
C
)
z
′
D
w
′
) = x log F +y log G+z log(I/pw
C
) +w log J
′
.
Letting R
def
= log F and γ denote A
x
′
B
y
′
(C/pw
C
)
z
′
D
w
′
, and using the fact that 'Server[E[F[G[I[J`
was output by the simulator in response to a Send
1
query, we may re-write these equations as
log µ = x +y log g
2
+z log h +w log(cd
β
) (3.6)
log ν
1
−r log E = x R +y Rlog g
2
+z R
′
log h +w Rlog(cd
β
) (3.7)
log ν
2
−log γ = x R +y Rlog g
2
+z R
′
log h +w R
′′
log(cd
β
), (3.8)
for some values R
′
, R
′′
∈ Z
q
such that R = R
′
and R = R
′′
. If R = 0 then R
′
, R
′′
= 0 and it is easy to
verify that (3.6)–(3.8) are linearly independent and not identically zero. If R = 0 one can similarly
verify that (3.6)–(3.8) are linearly independent and not identically zero. In either case, the joint
distribution of (K, sk
i
Client
, sk
j
Server
) is uniform independent of the remainder of the experiment.
In experiment P
8
, queries to the Send
0
oracle are handled diﬀerently. In particular, C is computed
as h
r
g
N+1
1
, where ¦1, . . . , N¦ is the dictionary of legal passwords.
Claim 3.9 Under the DDH assumption, [Adv
A,P8
(k) −Adv
A,P7
(k)[ ≤ ε(k) for some negligible func-
tion ε().
The proof exactly follows that of Claim 3.7. In particular, in responding to Send
2
queries, the
simulator never requires the value r to compute a session key: if msg-in is previously-used, the
session key (if it is assigned a value) is assigned a randomly-chosen value; if msg-in is new and
appears invalid (which can be veriﬁed using the decryption oracle), the session key (if it is assigned
a value) is assigned a random value; ﬁnally, if msg-in is new and appears valid (which can be veriﬁed
using the decryption oracle), the session key (if it is assigned a value) is assigned ∇.
The adversary’s view in experiment P
8
is independent of the passwords chosen by the simulator,
until one of the following occurs:
53
• The adversary queries Reveal(Client, i) or Test(Client, i), where the adversary had previously
queried Send
2
(Client, i, msg-in) and msg-in was new and appeared valid.
• The adversary queries Reveal(Server, j) or Test(Server, j), where the adversary had previously
queried Send
3
(Server, j, msg-in) and ﬁrst-msg-in was new and appeared valid.
The probability that one of these events occurs is therefore at most Q(k)/N, where Q(k) is the
number of on-line attacks made by A. The adversary succeeds when one of the above events occurs
or else by guessing the value of b. Assuming neither of the above events occur and the adversary
queries Test(U, i) where Π
i
U
is fresh and acc
i
U
= true, then sk
i
U
is randomly-distributed in G
independent of the rest of the experiment. Thus, the adversary’s probability of success in this case
is at most 1/2. Therefore
Pr
A,P8
[Succ] ≤ Q(k)/N +
1
2
(1 −
Q(k)
N
)
and the adversary’s advantage in experiment P
8
is at most Q(k)/N. Claims 3.1–3.9 show that
Adv
A,P0
(k) ≤ Q(k)/N +ε(k)
for some negligible function ε() and therefore the original P
0
is a secure, password-only, key-exchange
protocol.
Theorem 3.2 Assuming (1) the hardness of the DDH problem for groups G deﬁned by the output of
(; (2) the security of (/, Sign, Vrfy) as a one-time signature scheme; and (3) the security of UOWH
as a universal one-way hash family, the protocol of Figure 3.1 is a forward-secure, password-only,
key-exchange protocol.
Proof We refer to the formal speciﬁcation of the protocol in Figures 3.2–3.4. Here, however, the
adversary also has access to a Corrupt oracle as speciﬁed in Figure 3.8. As in the proof of Theorem 3.1,
Corrupt(Client, pw, Server) —
if Client / ∈ Client or Server / ∈ Server
return ⊥
if pw =⊥
pw
Server,Client
:= pw
return pw
Client
Figure 3.8: Speciﬁcation of the Corrupt oracle.
we imagine a simulator that runs the protocol for any adversary A. When the adversary completes
its execution and outputs a bit b
′
, the simulator can tell whether the adversary succeeds by checking
54
whether (1) a single Test query was made on instance Π
i
U
; (2) acc
i
U
was true at the time of the
Test query; (3) instance Π
i
U
is fs-fresh; and (4) b
′
= b. Success of the adversary is denoted by event
fsSucc. We refer to the real execution of the experiment as P
0
.
In experiment P
′
0
, the simulator interacts with the adversary as before except that the adversary
does not succeed when any of the following occur:
1. Any of g
1
, g
2
, h, c, d are not generators of G (i.e., they are equal to 1).
2. At any point during the experiment, a veriﬁcation key VK used by the simulator in responding
to a Send
0
query is repeated.
3. At any point during the experiment, the adversary forges a new, valid message/signature pair
for any veriﬁcation key used by the simulator in responding to a Send
0
query.
4. At any point during the experiment, a value β used by the simulator in responding to Send
1
queries is repeated.
5. At any point during the experiment, a value β used by the simulator in responding to a
Send
1
query (with msg-out = 'Server[E[F[G[I[J`) is equal to a value β used by the simulator
in responding to a Send
2
query (with msg-in = 'Server
′
[E
′
[F
′
[G
′
[I
′
[J
′
`) and furthermore
'Server[E[F[G[I` = 'Server
′
[E
′
[F
′
[G
′
[I
′
`.
Since the adversary, by deﬁnition, cannot succeed once any of these events occur, we assume that
the simulator immediately halts execution if any of these events occur. Using the same proof as for
Claim 3.1, it is clear that fsAdv
A,P0
(k) ≤ fsAdv
A,P
′
0
(k) +ε(k) for some negligible function ε().
In experiment P
1
, queries to the Execute oracle are handled diﬀerently. Upon receiving oracle
query Execute(Client, i, Server, j), the simulator checks whether pw
Client
= pw
Server,Client
. If so, the
values C and I used in responding to Execute queries are chosen independently at random from G
and the session keys are computed as
sk
i
Client
:= sk
j
Server
:= A
x2
B
y2
(C/pw
Server,Client
)
z2
D
w2
F
x1
G
y1
(I/pw
Client
)
z1
J
w1
.
On the other hand, if pw
Client
= pw
Server,Client
(i.e., as the result of a Corrupt oracle query), the
values C and I are chosen independently at random from G and the session keys are computed as
sk
i
Client
:= A
x2
B
y2
(C/pw
Client
)
z2
D
w2
F
x1
G
y1
(I/pw
Client
)
z1
J
w1
sk
j
Server
:= A
x2
B
y2
(C/pw
Server,Client
)
z2
D
w2
F
x1
G
y1
(I/pw
Server,Client
)
z1
J
w1
.
Using a similar proof as for Claim 3.2, it can be shown that, under the DDH assumption, we have
[fsAdv
A,P
′
0
(k) −fsAdv
A,P1
(k)[ ≤ ε(k) for some negligible function ε().
55
In experiment P
2
, queries to the Execute oracle are again handled diﬀerently. Upon receiving
query Execute(Client, i, Server, j), the simulator checks whether pw
Client
= pw
Server,Client
. If so,
sk
i
Client
is chosen randomly from G and sk
j
Server
is set equal to sk
i
Client
. Otherwise, both sk
i
Client
and sk
j
Server
are chosen independently at random from G.
Claim 3.10 [fsAdv
A,P1
(k) −fsAdv
A,P2
(k)[ ≤ ε(k) for some negligible function ε().
The claim follows from the negligible statistical diﬀerence between the distributions on the adver-
sary’s view in the two experiments. The case of pw
Client
= pw
Server,Client
exactly parallels the proof
of Claim 3.3. So, consider an invocation of the Execute oracle when pw
Client
= pw
Server,Client
(for
brevity, denote pw
Client
by pw
C
and pw
Server,Client
by pw
S,C
). We may write C = h
r
′
1
pw
C
= h
r
′′
1
pw
S,C
and I = h
r
′
2
pw
C
= h
r
′′
2
pw
S,C
where r
′
1
= r
′′
1
and r
′
2
= r
′′
2
. With all but negligible probability, we
also have r
1
= r
′
1
, r
′′
2
and r
2
= r
′
2
, r
′′
2
. Now, for any µ
1
, µ
2
, ν
1
, ν
2
∈ G and ﬁxing the random choices
for the remainder of experiment P
1
, the probability over choice of x
1
, y
1
, z
1
, w
1
, x
2
, y
2
, z
2
, w
2
that
E = µ
1
, K = µ
2
, sk
i
Client
= ν
1
, and sk
j
Server
= ν
2
is exactly the probability that
log
g1
µ
1
= x
2
+y
2
log
g1
g
2
+z
2
log
g1
h +w
2
log
g1
(cd
α
) (3.9)
log
g1
µ
2
= x
1
+y
1
log
g1
g
2
+z
1
log
g1
h +w
1
log
g1
(cd
β
) (3.10)
log
g1
ν
1
= x
1
r
2
+y
1
r
2
log
g1
g
2
+z
1
r
′
2
log
g1
h +w
1
r
2
log
g1
(cd
β
)
+x
2
r
1
+y
2
r
1
log
g1
g
2
+z
2
r
′
1
log
g1
h +w
2
r
1
log
g1
(cd
α
) (3.11)
log
g1
ν
2
= x
1
r
2
+y
1
r
2
log
g1
g
2
+z
1
r
′′
2
log
g1
h +w
1
r
2
log
g1
(cd
β
)
+x
2
r
1
+y
2
r
1
log
g1
g
2
+z
2
r
′′
1
log
g1
h +w
2
r
1
log
g1
(cd
α
). (3.12)
Since (3.9)–(3.12) are linearly independent and not identically zero when r
1
= r
′
1
, r
′′
2
and r
2
= r
′
2
, r
′′
2
,
the values E, K, sk
i
Client
, and sk
j
Server
are independently and uniformly distributed, independent of
the rest of the experiment.
In experiment P
3
, the simulator runs the modiﬁed initialization procedure shown in Figure 3.6,
with the values χ
1
, χ
2
, ξ
1
, ξ
2
, κ stored as before. Deﬁne the terms previously-used and new as in
the proof of Theorem 3.1. As before, a new msg-in for a query Send
2
(Client, i, 'Server[E[F[G[I[J`)
is said to appear valid only if F
χ1+βξ1
G
χ2+βξ2
= J and I/pw
Client
= F
κ
. Otherwise, it appears
invalid. A new msg-in for a query Send
1
(Server, j, 'Client[VK[A[B[C[D`) appears valid only if
A
χ1+αξ1
B
χ2+αξ2
= D and C/pw
Server,Client
= A
κ
, where the value of pw
Server,Client
is that at the
time of the Send
1
query (this is an important point to bear in mind, since the value of pw
Server,Client
may change as a result of Corrupt queries).
In experiment P
3
, queries to the Send
2
oracle are handled diﬀerently before any Corrupt queries
have been made (after a Corrupt query is made, the behavior of the Send
2
oracle is as in experiment
56
P
2
). Upon receiving query Send
2
(Client, i, msg-in), the simulator examines msg-in. If msg-in is new
and appears invalid, the query is answered as in experiment P
2
. If msg-in is new and appears valid,
the query is answered as in experiment P
2
but the simulator stores the value (∇, sk
i
Client
) as the
“session key”. If msg-in is previously-used, the simulator checks for the Send
1
query following which
msg-in was output (note that this query will be unique, since the experiment is aborted if a value
β used by the Send
1
oracle repeats). Say this query was Send
1
(Server, j, msg-in
′
). If msg-in
′
is new
and appears invalid, the Send
2
query is answered as in experiment P
2
. If msg-in
′
is new and appears
valid, the query is answered as in experiment P
2
but the simulator stores the value (∇, sk
i
Client
) for
sk
i
Client
.
Queries to the Send
3
oracle are also handled diﬀerently. Upon receiving a query of the form
Send
3
(Server, j, msg-in) and assuming used
j
Server
is true, the simulator ﬁrst checks when the query
Send
1
(Server, j, ﬁrst-msg-in) was made. If this Send
1
query was made after any Corrupt query was
made, the behavior of the Send
3
oracle is unchanged. Otherwise, the simulator checks ﬁrst-msg-in
and responds as in experiment P
2
unless ﬁrst-msg-in is new and appears valid. When ﬁrst-msg-
in is new and valid the query is answered as in experiment P
2
but the simulator stores the value
(∇, sk
j
Server
) as the “session key”.
The Test and Reveal oracle queries are also handled diﬀerently. If the adversary queries Test(U, i)
or Reveal(U, i) before any Corrupt queries have been made and the session key stored is of the form
(∇, sk
i
U
), the adversary is given ∇. If the Test or Reveal query is made after any Corrupt query
has been made, the adversary is given the value sk
i
U
. Test or Reveal queries where the session key
is not stored along with ∇ are answered as in experiment P
2
. Finally, the deﬁnition of success
is changed. If the adversary ever receives the value ∇ in response to a Reveal or Test query, the
adversary succeeds and the experiment is aborted (note that this can only occur before a Corrupt
query has been made). Otherwise, the adversary may succeed, as before, by correctly guessing the
bit b.
As in the proof of Theorem 3.1, we clearly have fsAdv
A,P2
(k) ≤ fsAdv
A,P3
(k) since the number
of ways the adversary can succeed is increased.
In general, the actions of the Send
2
oracle will be modiﬁed only when the Send
2
query is
made before any Corrupt queries have been made. Similarly, actions of the Send
3
oracle on query
Send
3
(Server, j, msg-in) will be modiﬁed only if the query Send
1
(Server, j, ﬁrst-msg-in) was made
before any Corrupt queries were made (if no Send
1
query of this form was made, the query to Send
3
simply returns ⊥). For clarity, however, we will still explicitly mention this condition when describing
the modiﬁed experiments.
In experiment P
4
, queries to the Send
3
oracle are handled diﬀerently. First, whenever the sim-
57
ulator responds to a Send
2
query, the simulator stores the values (K, β, x, y, z, w), where K =
g
x
1
g
y
2
h
z
(cd
β
)
w
. Upon receiving query Send
3
(Server, j, 'K[Sig`) (as before, this assumes that query
Send
1
(Server, j, ﬁrst-msg-in) was made before any Corrupt queries), the simulator checks the value
of ﬁrst-msg-in = 'Client[VK[A[B[C[D` (if ﬁrst-msg-in is not deﬁned the query to Send
3
simply re-
turns ⊥ as in experiment P
3
). If ﬁrst-msg-in is new, the query is answered as in experiment P
3
. If
ﬁrst-msg-in is previously-used and Vrfy
VK
(β[K, Sig) = 0, the query is answered as in experiment P
3
and the session key is not assigned a value. If ﬁrst-msg-in is previously-used, Vrfy
VK
(β[K, Sig) = 1,
and the experiment is not aborted, the simulator ﬁrst checks whether there exists an i such that
sid
i
Client
= sid
j
Server
(if such an i exists, it must be unique since veriﬁcation keys VK are not re-
peated during Send
0
queries). If so, sk
j
Server
(if it is assigned a value at all) is assigned the value
sk
i
Client
. Otherwise, let ﬁrst-msg-out = 'Server[E[F[G[I[J`. The simulator must have stored values
x
′
, y
′
, z
′
, w
′
such that K = g
x
′
1
g
y
′
2
h
z
′
(cd
β
)
w
′
(this is true since the experiment is aborted if Sig is a
valid signature on β[K that was not output by the simulator following a Send
2
query). The session
key (assuming it is assigned a value at all) is assigned the value:
sk
j
Server
:= A
x
B
y
(C/pw)
z
D
w
F
x
′
G
y
′
(I/pw)
z
′
J
w
′
.
Claim 3.11 fsAdv
A,P4
(k) = fsAdv
A,P3
(k).
The distribution on the adversary’s view is identical in experiments P
3
and P
4
; the proof is as for
Claim 3.5. It is crucial here that the value pw
Server,Client
used when responding to a Send
1
query is
stored as part of the state and thus the same value is used subsequently when responding to a Send
3
query (cf. Figure 3.4). If this were not the case, the value of pw
Server,Client
could change (as a result
of a Corrupt query) sometime between the Send
1
and Send
3
queries.
In experiment P
5
, queries Send
3
(Server, j, msg-in) are handled diﬀerently (again, this assumes
that the query Send
1
(Server, j, ﬁrst-msg-in) was made before any Corrupt queries). Upon receiving
query Send
3
(Server, j, 'K[Sig`), the simulator checks the value of ﬁrst-msg-in (if ﬁrst-msg-in is not
deﬁned the simulator returns ⊥ as in experiment P
4
). If ﬁrst-msg-in is new and appears invalid and
the session key is to be assigned a value, the session key is assigned a value randomly chosen in G.
Otherwise, the query is answered as in experiment P
4
.
Claim 3.12 fsAdv
A,P5
(k) = fsAdv
A,P4
(k).
The proof exactly follows the proof of Claim 3.6. Again, it is crucial that the same value pw
Server,Client
is used during both the Send
1
and Send
3
queries for a given instance (this is achieved by storing
pw
Server,Client
as part of the state).
58
In experiment P
6
, queries to the Send
1
oracle are handled diﬀerently when a Send
1
query is made
before any Corrupt queries (Send
1
queries are responded to as in experiment P
5
when they are made
after a Corrupt query). Upon receiving query Send
1
(Server, j, msg-in), if msg-in is new and appears
valid the query is answered as before. Otherwise, component I is computed as h
r
g
N+1
1
, where the
dictionary of legal passwords is ¦1, . . . , N¦; note that g
N+1
1
represents an invalid password since
N < q −1.
Claim 3.13 Under the DDH assumption, [fsAdv
A,P5
(k) −fsAdv
A,P6
(k)[ ≤ ε(k), for some negligible
function ε().
The proof of this claim exactly follows the proof of Claim 3.7. In particular, the claim follows from
the non-malleability of the commitment scheme used. When a query Send
1
(Server, j, ﬁrst-msg-in) is
made before any Corrupt queries and ﬁrst-msg-in is not both new and valid, the simulator will not
require r in order to respond to the (subsequent) query Send
3
(Server, j, msg-in) in case this query
is ever made.
In contrast to the basic case when no Corrupt queries are allowed, the simulator does require r
in case ﬁrst-msg-in is both new and appears valid. The reason is the following: Assume the ad-
versary queries Send
1
(Server, j, 'Client[VK[A[B[C[D`) (before any Corrupt queries have been made)
where 'Client[VK[A[B[C[D` is new and appears valid. Assume the adversary subsequently learns
pw
Server,Client
= pw from a Corrupt query. The adversary might then query Send
3
(Server, j, 'K[Sig`)
followed by Reveal(Server, j). In this case, the simulator must give the adversary the correct session
key sk
j
Server
— but this will be impossible without r.
In experiment P
7
, queries to the Send
2
oracle are handled diﬀerently (when they are made before
any Corrupt queries). Whenever msg-in is new and appears invalid, the session key (if it is assigned a
value at all) is assigned a value chosen randomly from G. If msg-in is previously-used, the simulator
checks for the Send
1
query after which msg-in was output (note that this query will be unique,
since values β used by the Send
1
oracle do not repeat). Say this query was Send
1
(Server, j, msg-in
′
).
If msg-in
′
is new and appears valid, the Send
2
query is answered as before. If msg-in
′
is new
and appears invalid, the session key (if it is assigned a value at all) is assigned a value chosen
randomly from G. Queries Send
3
(Server, j, msg-in) are also handled diﬀerently (assuming query
Send
1
(Server, j, ﬁrst-msg-in) was made before any Corrupt queries). If ﬁrst-msg-in is previously-
used, Vrfy
VK
(β[K, Sig) = 1, and there does not exist an i such that sid
i
Client
= sid
j
Server
, the session
key is assigned a value chosen randomly from G.
Claim 3.14 fsAdv
A,P7
(k) = fsAdv
A,P6
(k).
The proof exactly follows that of Claim 3.8.
59
Let fsSucc1 denote the event that the adversary succeeds by receiving a value ∇ following a Test
or Reveal query (note that this event can only occur before any Corrupt queries have been made).
We have
Pr
A,P7
[fsSucc] = Pr
A,P7
[fsSucc1] + Pr
A,P7
[fsSucc[fsSucc1] Pr
A,P7
[fsSucc1].
Event fsSucc ∧ fsSucc1 can occur in one of two ways (by deﬁnition of fs-freshness): either (1) the
adversary queries Test(U, i) before any Corrupt queries and does not receive ∇ in return; or (2)
the adversary queries Test(U, i) after a Corrupt query and acc
i
U
= true but the adversary has
never queried Send
n
(U, i, M) for any n, M. In either case, sk
i
U
is randomly chosen from G indepen-
dent of the remainder of the experiment; therefore (assuming Pr
A,P7
[fsSucc1] = 0) we must have
Pr
A,P7
[fsSucc[fsSucc1] = 1/2. In other words
Pr
A,P7
[fsSucc] = 1/2 + 1/2 Pr
A,P7
[fsSucc1],
where we assume Pr
A,P7
[fsSucc1] = 0. Below, we give an upper bound on Pr
A,P7
[fsSucc1] which, in
particular, will show that Pr
A,P7
[fsSucc1] = 0.
We deﬁne experiment P
′
7
in which the adversary succeeds only if it receives a value ∇ in response
to a Reveal or Test query. Clearly Pr
A,P
′
7
[fsSucc] = Pr
A,P7
[fsSucc1] by deﬁnition of event fsSucc1.
Since the adversary cannot succeed in experiment P
′
7
once a Corrupt query is made, the simulator
aborts the experiment if this is ever the case. For this reason, whenever the simulator would
previously store values (∇, sk) as a “session key” (with sk being returned only in response to a Test
or Reveal query after a Corrupt query), the simulator now need store only ∇.
In experiment P
8
, queries to the Send
1
oracle are handled diﬀerently. Now, the simulator always
computes I as h
r
g
N+1
1
(i.e., even when msg-in is new and valid). That [fsAdv
A,P8
(k)−fsAdv
A,P
′
7
(k)[ ≤
ε(k) for some negligible function ε() (under the DDH assumption) follows from a proof similar to
that of Claim 3.7 (see also Claim 3.13). A key point is that when msg-in is new and appears valid,
the simulator no longer need worry about simulating the adversary’s view following a Corrupt query.
In experiment P
9
, queries to the Send
0
are handled diﬀerently. Now, the simulator always
computes C as h
r
g
N+1
1
. That [fsAdv
A,P9
(k) − fsAdv
A,P8
(k)[ ≤ ε(k) for some negligible function
ε() (under the DDH assumption) follows from a proof similar to that of Claim 3.9. In particular,
since session keys computed during a Send
2
query are always either ∇ or are chosen randomly from
G, the value r is not needed by the simulator and hence we can use the simulator to break the
Cramer-Shoup encryption scheme as in the proofs of Claims 3.7, 3.9, and 3.13.
The adversary’s view in experiment P
9
is independent of the passwords chosen by the simulator
until the adversary receives ∇ in response to a Reveal or Test query (in which case the adversary
60
succeeds). Thus, the probability that the adversary receives ∇ (which is exactly Pr
A,P9
[fsSucc]) is
at most Q(k)/N. When Q(k) < N (which is the only interesting case), we have Pr
A,P7
[fsSucc1] < 1
for k large enough. Therefore, for large enough values of k and for some negligible function ε(), we
have
Pr
A,P0
[fsSucc] ≤ Pr
A,P7
[fsSucc] +ε(k)
≤ 1/2 + 1/2 Pr
A,P7
[fsSucc1] +ε(k)
≤ 1/2 +
Q(k)
2N
+ε(k)
and fsAdv
A,P0
(k) ≤ Q(k)/N +2 ε(k). In other words, the original P
0
is a forward-secure, password-
only, key-exchange protocol.
61
Chapter 4
Non-Interactive and
Non-Malleable Commitment
4.1 Introduction
Consider a setting in which two parties must each choose a course of action (by declaring their choice
to a receiver), yet neither party should have the advantage of moving second and thereby basing
their choice on the other player’s selection. How can this be achieved? There are several natural
solutions to this problem. One is to force both players to announce their choices to the receiver
at exactly the same time. Clearly, this will achieve the desired result if absolute simultaneity can
be achieved; in practice, however, guaranteeing this level of synchrony is diﬃcult if not impossible.
Another suggestion is to involve an “escrow agent” to whom each party communicates his choice.
This escrow agent should not reveal either player’s choice to the receiver until both players have
communicated with him. Although this approach accomplishes the desired task, it involves an
additional party who must be highly trusted by all others involved. This solution also places a
heavy burden on the agent in case many parties want to use his services at the same time.
A particularly simple and elegant solution is to have both parties “commit” to their choice
such that the following holds: (1) a commitment should reveal no information about the choice
being committed to, yet (2) once a commitment is made, the committing party should be unable
to change their committed value. A straightforward way to achieve this is to have a committing
party write their choice on a piece of paper, seal it inside an envelope, and send the envelope to
the receiver. This approach satisﬁes both requirements given above: the second party (even if he
observes all communication between the other party and the receiver) sees only the envelope and
gets no information about the contents inside; furthermore, the ﬁrst party cannot change what is
inside the envelope once it is sealed.
The preceding solution works when the parties and the receiver are physically close (and can
62
o (input m ∈ Z
q
)
Commitment phase:
g, h ←G
∗
g, h

r ←Z
q
com := g
m
h
r
com
-
{
Decommitment phase:
m, r
-
Verify: com
?
= g
m
h
r
Figure 4.1: The Pedersen commitment scheme.
therefore quickly pass an envelope back and forth), but is not feasible in many other situations.
A commitment protocol may be viewed as the cryptographic implementation of a secure envelope.
Here, we can guarantee that the committed value is not revealed (secrecy) and that the committing
party cannot change his mind (binding) under assumptions about the computational power of the
parties.
An example will be instructive. In Figure 4.1, we illustrate the Pedersen commitment scheme
[103] whose security is based on the hardness of computing discrete logarithms in group G of prime
order q. First, the receiver { chooses two random generators g, h ∈ G and sends these to the sender
o. To commit to a message m ∈ Z
q
, o chooses a random value r ∈ Z
q
, computes com = g
m
h
r
,
and sends com to {. To decommit, the sender simply reveals m, r. Note that com is uniformly
distributed in G, independently of m, and therefore the commitment reveals no information about
the committed message. On the other hand, assuming the hardness of computing discrete logarithms
in G, the Pedersen scheme is binding. To see this, note that given two legal decommitments 'm, r`
and 'm
′
, r
′
` to com, we have g
m
h
r
= g
m
′
h
r
′
. Thus, we may compute g
m−m
′
= h
r
′
−r
and log
g
h =
(m−m
′
)/(r
′
−r) mod q.
Two types of commitment schemes are primarily considered in the literature: perfectly-binding
and perfectly-hiding (following [62] we refer to the former as standard and the latter as perfect ). In
a standard commitment scheme, each commitment is information-theoretically linked to only one
possible (legal) decommitment value; on the other hand, the secrecy of the commitment is guaranteed
only with respect to a computationally-bounded receiver. In a perfect commitment scheme, the
secrecy of the commitment is information-theoretic while the binding property guarantees only that
63
o (input m ∈ Z
q
)
Commitment phase:
g, h ←G
∗
g, h

g, h

r ←Z
q
com := g
m
h
r
com
-
g com
-
{
Decommitment phase:
m, r
-
m+ 1, r
-
Verify: g com
?
= g
m+1
h
r
Figure 4.2: Man-in-the-middle attack on the Pedersen commitment scheme.
a computationally-bounded sender cannot ﬁnd a commitment which can be opened in two possible
ways. The type of commitment scheme to be used depends on the application; it may also depend on
assumptions regarding the computational power of the participants. Furthermore, in some protocols
(e.g., zero-knowledge proofs) certain commitments are never opened; information-theoretic privacy
ensures that the committed data will remain hidden indeﬁnitely.
Commitment protocols have become one of the most fundamental cryptographic primitives, and
are used as sub-protocols in such applications as zero-knowledge proofs [67, 62], secure multi-party
computation [66], and many others. Commitment protocols can also be used directly, for example,
in remote (electronic) bidding. In this setting, parties bid by committing to a value; once bidding is
complete, parties reveal their bids by decommitting.
In many situations, however, the secrecy and binding properties outlined previously do not fully
capture everything one might expect from a secure envelope. For example, we do not expect a party
to be able to commit to a value which is related in any way to a previously committed (and as yet
unopened) value. Schemes in which commitment to a related value is possible are called malleable;
schemes in which this is impossible are called non-malleable. As an illustration, in the bidding
scenario it is unacceptable if one party can generate a valid commitment to m + 1 upon viewing a
commitment to m. Note that the value of the original commitment may remain unknown (and thus
secrecy need not be violated); in fact, the second party may only be able to decommit his bid after
viewing a decommitment of the ﬁrst.
Unfortunately, most known commitment protocols are easily susceptible to these types of man-
in-the-middle attacks. For example, Figure 4.2 demonstrates the malleability of the Pedersen com-
64
mitment scheme. Here, the adversary changes the commitment com of o to com
′
= g com. At this
point, the adversary has no idea what o has committed to, nor what he himself has committed to.
In fact, the adversary will only be able to decommit after viewing the decommitment of o. When
o decommits to m, the adversary simply decommits to m
′
= m + 1. The receiver cannot even tell
that a man-in-the-middle attack is taking place.
1
One natural measure of eﬃciency for a commitment protocol is the communication complexity.
The number of rounds is one measure of the communication complexity. Optimally, the commitment
phase of a commitment protocol should consist of a single message from the sender to the receiver;
such schemes are termed non-interactive. A second measure of the communication complexity is the
bit complexity (i.e., size) of a commitment. This is particularly important when committing to a
very large message such as the contents of a (large) database. Unfortunately, standard commitment
schemes (even malleable ones) require commitment size at least [M[ + ω(log k), where [M[ is the
message size and k is the security parameter. Perfect commitment schemes, on the other hand, oﬀer
the opportunity to achieve much shorter commitment lengths.
4.1.1 Previous Work
The commitment primitive has been extensively studied. Standard commitment has been shown
to exist if and only if one-way functions exist [96, 77]. A perfect commitment scheme has been
constructed assuming the existence of one-way permutations [97]. Both schemes have been designed
in the interactive model (where no public information is available to the parties); the former, how-
ever, may be adapted to run in the public-parameters model (cf. Section 2.1.1). Eﬃcient perfect
commitment protocols, based on speciﬁc number-theoretic assumptions, are also known [103, 100].
Non-malleability of commitments was ﬁrst explicitly considered by Dolev, Dwork, and Naor
[44]. They also provided the ﬁrst construction of a standard commitment scheme which is provably
non-malleable. Although their protocol is constructed from the minimal assumption of a one-way
function (in particular, without assuming any public parameters), it requires a poly-logarithmic
number of rounds of interaction.
2
Assuming a public random string available to all participants,
Di Crescenzo, Ishai, and Ostrovsky [40] construct the ﬁrst non-interactive, non-malleable standard
commitment scheme. Interestingly, their construction can be modiﬁed to give a non-interactive,
non-malleable perfect commitment scheme. Unfortunately, the resulting commitments are large (i.e.,
O([M[ k)), thus motivating the search for more eﬃcient protocols. Furthermore, their protocol is
1
In the present example, a receiver who obtains commitments and decommitments from both S and the adversary
can tell that something unusual happened since the decommitments are correlated (for example, they both use the
same r). However, it is possible for an adversary to prevent this detection by re-randomizing his commitment.
2
Furthermore, their protocol allows an adversary to generate a diﬀerent commitment to an identical value (unless
user identities are assumed). The protocols we present do not have this drawback.
65
not computationally practical.
Eﬃcient non-malleable commitment schemes, based on stronger (but standard) assumptions,
have been given by Fischlin and Fischlin [58]. Like the construction of [40], these protocols require
publicly-available parameters generated by a trusted party (in some cases this can be weakened to the
assumption of a public random string). They describe non-malleable perfect commitment schemes
based on either the discrete logarithm or RSA assumptions. Though eﬃcient, these protocols require
interaction between the sender and receiver.
Subsequent to the present work, a deﬁnition of “universally composable” commitment protocols
(which implies, in particular, non-malleability) was introduced, and a provably-secure construction
satisfying this deﬁnition was given in the common-random-string model [28]. Unfortunately, the
given protocol is ineﬃcient for commitment to more than a single bit. Other subsequent work,
building on the paradigm described here, has demonstrated an eﬃcient non-malleable commitment
scheme based on the factoring assumption [59].
4.1.2 Our Contributions
We present the ﬁrst eﬃcient (in both computation and communication) constructions of non-
interactive, non-malleable, perfect commitment schemes. We work in the setting in which public
parameters are available to all participants (our discrete logarithm construction can be implemented
in the public random string model using standard techniques). Previous constructions are either for
the case of standard commitment [44, 40] or require interaction [44, 58]. Our constructions are based
on the discrete logarithm or the RSA assumptions, and allow eﬃcient, perfectly-hiding commitment
to arbitrarily-large messages. The size of the resulting commitment is essentially optimal. The
schemes described in [58], while able to handle large messages, require modiﬁcations which render
them less eﬃcient and result in statistical secrecy.
We also discuss the case of non-interactive, non-malleable, standard commitment and prove
secure a construction based on trapdoor permutations that achieves commitment size [M[ +poly(k).
The large commitment size of this construction (though near-optimal for standard commitment)
serves as motivation for our consideration of perfect commitment schemes. Indeed, for arbitrarily-
large messages, our perfect commitment schemes yield commitments of size O(k), where k is the
security parameter (the commitment size is further improved in Section 4.5). All our schemes require
only poly(k) bits of public information, independent of the size of the committed message.
66
4.2 Deﬁnitions
Commitment schemes. A non-interactive commitment scheme
3
[40] in the public-parameters
model is deﬁned by a triple of probabilistic, polynomial-time algorithms (T T {, o, {) which describe
a two-phase protocol between a sender o and a receiver { such that the following is true. In the
ﬁrst phase (the commitment phase), the sender o, given a public string σ output by a trusted
third party T T {, commits to a message m by computing a pair of strings (com, dec) and sending
the commitment com to receiver {. Given only σ and com, the receiver cannot determine any
information about m (this is the hiding property). In the second phase (the decommitment phase)
o reveals the decommitment dec to { and { checks whether the decommitment is valid. If it is
not, { outputs a special string ⊥, meaning that he rejects the decommitment from o; otherwise, {
can eﬃciently compute the message m and is convinced that m was indeed chosen by o in the ﬁrst
phase. It should be infeasible for o to generate a commitment which o can later decommit in more
than one possible way (this is the binding property).
In a perfect commitment scheme, the committed message is hidden from the receiver in an
information-theoretic sense. Thus, even an inﬁnitely-powerful receiver cannot determine the message
to which o has committed at the end of the ﬁrst phase. On the other hand, the binding property
only holds with respect to a computationally-bounded sender. Thus, a given commitment has
possible (legal) decommitments to many messages, but a polynomial-time sender cannot explicitly
ﬁnd more than one such decommitment. In contrast, a standard commitment scheme prevents
even an inﬁnitely-powerful sender from decommitting a commitment in more than one possible way;
however, at the end of the commitment phase the message is hidden only from a computationally-
bounded receiver. Formal deﬁnitions follow:
Deﬁnition 4.1 Let (T T {, o, {) be a triple of probabilistic, polynomial time algorithms, k a security
parameter, and ¦´
σ
¦
σ∈T T P(1
k
),k∈N
a collection of message spaces such that membership in ´
σ
is
eﬃciently testable given σ output by T T {(1
k
). We say that (T T {, o, {) is a non-interactive, perfect
(resp. standard) commitment scheme over ¦´
σ
¦ if the following conditions hold:
1. (Meaningfulness) For all k ∈ N, all σ output by T T {(1
k
), and all m ∈ ´
σ
:
Pr[(com, dec) ← o(σ, m); m
′
← {(σ, com, dec) : m
′
= m] = 1.
2. (Perfect (resp. computational) secrecy) For all computationally-unbounded (resp. ppt)
distinguishing algorithms D, the following is identically zero (resp. negligible in k):

σ ← T T {(1
k
); (com, dec
1
, dec
2
) ← o
′
(σ); m
1
← {(σ, com, dec
1
);
m
2
← {(σ, com, dec
2
) : m
1
=⊥ ∧ m
2
=⊥ ∧ m
1
= m
2
] .
Note. The symbol ⊥ is reserved for an invalid decommitment (which includes a refusal to decom-
mit). In particular, for all σ we have ⊥ / ∈ ´
σ
.
Non-malleable commitment schemes. Two deﬁnitions of non-malleable commitment have ap-
peared in the literature, both seeking to capture the following intuition of security: for any adversary
who, after viewing a commitment to m, produces a commitment to a value m
′
which bears some
relation to m, there exists a simulator performing at least as well in producing an m
′
related to m
but without viewing a commitment to m (and thus having no information about the value of m).
The diﬀerence between the two deﬁnitions lies in what it means for an adversary to“produce a com-
mitment”. In the original deﬁnition [44] (non-malleability with respect to commitment ), generating
a valid commitment to m
′
is suﬃcient. However, this deﬁnition does not apply to perfectly-hiding
commitment schemes since for such schemes the value committed to by a commitment string is not
well-deﬁned. In the deﬁnition of [40, 58] (non-malleability with respect to opening), the adversary
must also give a valid decommitment to m
′
after viewing the decommitment to m. Note that in the
case of standard commitment, non-malleability with respect to commitment is a stronger notion of
security.
Deﬁnition 4.2 A non-interactive commitment scheme (T T {, o, {) over ¦´
σ
¦ is ε-non-malleable
with respect to opening if, for all ε > 0 and every ppt algorithm /, there exists a simulator /
′
run-
ning in probabilistic, poly(k, 1/ε) time, such that for all polynomial-time computable, valid relations
R (see note below), we have:
Succ
NM
A,R
(k) −
¯
Succ
A
′
,R
(k) ≤ ε + negl(k),
for some negligible function negl(); where:
Succ
NM
A,R
(k)
def
=
Pr

.
(T represents an eﬃciently sampleable distribution over ´
σ
.)
Note. The deﬁnition of security above allows the simulator to do arbitrarily better than the adver-
sary. The technical reason for this is that the adversary may simply refuse to decommit, even when
it would have otherwise succeeded. In any case, if a simulator who has no information about m
1
can do better than an adversary who gets to see a commitment to m
1
, the scheme still satisﬁes our
intuitive notion of non-malleability.
Valid relations. Typically, we view a relation R as being deﬁned on pairs of messages. However,
there are a number of subtleties which need to be addressed. First, the adversary /should be allowed
to choose the distribution T, possibly depending upon the public parameters σ. Any reasonable
deﬁnition of security thus requires the simulator to output a distribution as well. The simulator,
however, must be prevented from choosing some “trivial” distribution for which it can always succeed.
To handle this, we allow the relation R to take the public parameters σ and (a description of) a
distribution T as additional parameters. To ensure that the distribution is deﬁned with respect to
the correct message space, we require that a valid relation satisfy R(σ, T, ∗, ∗) = 0 if T is not a
distribution over ´
σ
. Furthermore, we require R(σ, ∗, m
1
, m
2
) = 0 if m
1
/ ∈ ´
σ
or m
2
/ ∈ ´
σ
. In
particular, this implies that R(σ, T, m, ⊥) = R(σ, T, ⊥, m) = 0 for all σ, T, m.
We now turn to the deﬁnition of non-malleability with respect to commitment. For any standard
commitment scheme, we may deﬁne the function Decommit(σ, ) which takes a commitment as its
second argument and returns the (unique) committed value in ´
σ
(or ⊥, if the commitment is
invalid). The perfect binding of the scheme implies that the function is well-deﬁned (although not
polynomial-time computable) except with negligible probability over choice of σ.
Deﬁnition 4.3 A non-interactive, standard commitment scheme (T T {, o, {) over message space
¦´
σ
¦ is non-malleable with respect to commitment if, for every ppt algorithm /, there exists a ppt
simulator /
′
, such that for all polynomial-time, valid relations R (see note above), we have:

σ ← T T {(1
k
); (T, s) ← /
0
(σ); m
1
← T;
(com
1
, dec
1
) ← o(σ, m
1
); com
2
← /(σ, com
1
, s);
m
2
= Decommit(σ, com
2
) : com
1
= com
2
∧ R(σ, T, m
1
, m
2
) = 1]
and
¯
Succ
A
′
,R
(k) is deﬁned as above.
Equivocable commitment schemes. Our constructions of non-malleable, perfect commitment
schemes use equivocable commitment schemes [2] as a building block; such schemes have been used
previously in designing non-malleable commitment protocols [40]. Informally, an equivocable com-
mitment scheme in the public-parameter model is one for which there exists an eﬃcient equivocation
algorithm Equiv, substituting for the trusted third party, which outputs public parameters σ and
a commitment such that: (a) the distribution of σ, the commitment, and a decommitment to any
message is exactly equivalent to their distribution in a real execution of the protocol; and (b) the
commitment can be opened by Equiv in more than one possible way. We give a formal deﬁnition for
the case of perfect commitment.
Deﬁnition 4.4 A non-interactive, perfect commitment scheme (T T {, o, {) over message space
¦´
σ
¦ is perfectly equivocable if there exists a probabilistic, polynomial time equivocable commitment
generator Equiv such that:
1. Equiv
1
(1
k
) outputs (σ, com, s) (where s represents state information).
2. For all k ∈ N, the following distributions are equivalent:
¦σ ← T T {(1
k
); m ← ´
σ
; (com, dec) ← o(σ, m) : (σ, com, dec, m)¦
¦(σ, com, s) ← Equiv
1
(1
k
); m ← ´
σ
; dec ← Equiv
2
(s, m) : (σ, com, dec, m)¦.
In particular, for all k ∈ N, all (σ, com, s) output by Equiv
1
(1
k
), all m ∈ ´
σ
, and all dec output by
Equiv
2
(s, m) we have {(σ, com, dec) = m.
4.3 Non-Malleable Standard Commitment
We ﬁrst examine the case of non-interactive, standard commitment. Note that the size of a standard
commitment (even for malleable schemes) must be at least [M[ +ω(log k), where [M[ is the message
length and k is the security parameter. Perfect binding implies that the size must be at least [M[,
and semantic security requires that each message have 2
ω(log k)
= ω(poly(k)) possible commitments
associated with it.
70
The theorem below indicates that we can achieve roughly this bound for non-interactive, non-
malleable standard commitment, assuming the existence of trapdoor permutations
4
(in the model
with public parameters). The key realization is that a non-malleable public-key encryption scheme
can also be used as a non-malleable standard commitment scheme. This connection between non-
malleable public-key encryption and non-malleable commitment seems not to have been noticed
before. Following Blum and Goldwasser [21, 68] (who consider the case of semantic security for
public-key encryption), we construct a communication-eﬃcient, non-malleable standard commitment
scheme from the following components: ﬁrst, we use a public-key encryption scheme (Gen, c, T) that
is indistinguishable under an adaptive chosen-ciphertext attack (and hence non-malleable) [44, 6, 16].
Such a scheme can be based on any family of trapdoor permutations [44, 110, 38]. Next, we use a
symmetric-key cryptosystem (/, c
∗
, T
∗
) which is indistinguishable under adaptive chosen-ciphertext
attack; this can be based on any one-way function [44]. The commitment scheme works as follows:
public parameters σ consist of a public key pk for the public-key cryptosystem. Commitment is
done by choosing a random secret key K for the symmetric-key cryptosystem, encrypting K using
the public-key cryptosystem, and then encrypting the committed message using the symmetric-key
cryptosystem and key K. A commitment to M is thus
c
pk
(K) ◦ c
∗
K
(M). (4.1)
Decommitment consists of revealing M and the random bits used to form the commitment. Com-
mitment veriﬁcation is done in the obvious way. Note that the commitment is actually a public-key
encryption of the message M, although the receiver does not have the associated secret key (indeed,
no party has it because σ was generated by T T {). Furthermore, [σ[ = [pk[ = poly(k) independent
of the size of the committed message. We now prove the following:
Theorem 4.1 Construction (4.1) is a non-interactive, standard commitment scheme (in the public-
parameters model) that is non-malleable with respect to commitment and has commitment size
[M[ + poly(k), where [M[ is the size of the committed message and k is the security parameter.
Furthermore, (4.1) may be based on the existence of trapdoor permutations.
Proof First note that for (4.1) to be perfectly binding we require that the decryption algorithms
for both the public-key and symmetric-key cryptosystems have zero probability of decryption error.
This is achieved, in particular, by the public-key and symmetric-key cryptosystems given in [44],
which may be based on any family of trapdoor permutations. Thus, revealing the randomness used
to generate the commitment perfectly binds the sender to the message.
4
Recall that [40] achieves non-interactive, non-malleable, standard commitment assuming the existence of one-way
functions. However, their scheme requires commitment size O(k · |M|).
71
A proof of non-malleability with respect to commitment will immediately imply that the scheme
is computationally hiding (this has been noted previously for the case of encryption [6, 16] and it
is clear that a similar result holds for the case of commitment). Since any commitment to M using
the scheme outlined above may also be viewed as an encryption of M under public key pk = σ, if we
can prove that (4.1) constitutes a non-malleable public-key encryption scheme, we are done. Using
the results of [6], it suﬃces to prove that (4.1) is secure under adaptive chosen-ciphertext attack.
Consider an adversary /attacking construction (4.1) under an adaptive chosen-ciphertext attack.
Deﬁne adversary B using / as a black box to attack c under an adaptive chosen-ciphertext attack
as follows (
¯
T
sk
() denotes the decryption oracle for hybrid scheme (4.1)):
Algorithm B
D
sk
(·)
1
(pk)
(M
0
, M
1
, s) ← /

.
The crux of the proof is to note that when T
sk
(y) = K, oracle
¯
T
sk,y→K
() is equivalent to the real
decryption oracle
¯
T
sk
() for the hybrid encryption scheme. With this in mind, we may re-write
(4.2) and (4.3) as Adv
B,E
(k) = 1/2 [p
0,real
(k) − p
0,fake
(k) + p
1,fake
(k) − p
1,real
(k)[ and Adv
C,E
∗(k) =
[p
0,fake
(k) −p
1,fake
(k)[. Then:
Adv
A
(k)
def
= [p
0,real
(k) −p
1,real
(k)[
= [p
0,real
(k) −p
1,real
(k) +p
0,fake
(k) −p
1,fake
(k) −p
0,fake
(k) +p
1,fake
(k)[
≤ [p
0,real
(k) −p
1,real
(k) −p
0,fake
(k) +p
1,fake
(k)[ +[p
0,fake
(k) −p
1,fake
(k)[
= 2 Adv
B,E
(k) + Adv
C,E
∗(k)
≤ 2 ε
1
(k) +ε
2
(k),
and this last quantity is negligible.
To complete the proof, we note that [K[ = poly(k) and therefore [c
pk
(K)[ = poly(k). Further-
more, we can achieve [c
∗
K
(M)[ = [M[ + k using a stream cipher and a secure mac (see [44]). The
73
total length of the commitment given by (4.1) is then [M[ + poly(k).
This theorem immediately implies the security (under the decisional Diﬃe-Hellman assumption)
of construction (4.1) when using the eﬃcient public-key cryptosystem of [36] and any adaptive
chosen-ciphertext-secure symmetric-key cryptosystem (/, c
∗
, T
∗
). We note that the security re-
quirements for the public- and private-key encryption schemes can be relaxed: (Gen, c, T) is only
required to be non-malleable under a chosen-plaintext attack (i.e., secure in the sense of NM-CPA)
and (/, c
∗
, T
∗
) need only be indistinguishable under a P0 plaintext attack and an adaptive chosen-
ciphertext attack (i.e., secure in the sense of IND-P0-C2); see [6, 84] for formal deﬁnitions. This
is so because (4.1) is a non-malleable commitment scheme whenever it is secure in the sense of
NM-CPA when viewed as a public-key encryption scheme (in the proof above, we show that (4.1) is
secure in the sense of NM-CCA2). This allows for much greater eﬃciency since NM-CPA public-key
cryptosystems can be constructed more eﬃciently than IND-CCA2 schemes [47] and IND-P0-C2
symmetric-key schemes may be deterministic.
We remark that the result in the theorem applies in the public random string model when
chosen-ciphertext-secure dense [39] public-key encryption schemes are used.
4.4 Non-Malleable Perfect Commitment
The computationally-hiding commitment scheme presented above achieves commitment size [M[ +
poly(k). This cannot be improved very much, since computationally-hiding commitments have size at
least [M[. In this section (see also Section 4.5) we present perfectly-hiding commitment schemes that
improve signiﬁcantly on the commitment length, achieving commitment size O(k) for arbitrarily-
large messages.
Both of our perfectly-hiding commitment schemes build on the paradigm established in [40],
with modiﬁcations which substantially improve the eﬃciency. A commitment consists of three
components 'A, B, Tag`. The ﬁrst component A is a commitment to a random key r
1
for a one-time
message authentication code (mac). The second component B contains the actual commitment
to the message m, using public parameters which depend upon the ﬁrst component A. Finally,
Tag = mac
r1
(B). An adversary who wishes to generate a commitment to a related value has two
choices: he can either re-use A or use a diﬀerent A
′
. If he re-uses A, with high probability he will
be unable to generate a correct Tag for a diﬀerent B
′
, since he does not know the value r
1
. On the
other hand, if he uses a diﬀerent A
′
, the public parameters he is forced to use for his commitment
B
′
will be diﬀerent from those used for the original commitment; thus, the adversary will be able
to decommit in only one way, regardless of how the original B is decommitted. In particular, if
it is possible for a simulator to equivocate B for a particular choice of A, an adversary who uses
74
Public: G, g1, g2, g3; H : G → Zq
o (input m ∈ Z
q
)
Commitment phase:
r
1
, r
2
, r
3
←Z
q
A := g
r1
1
g
r2
3
; α := H(A)
B := (g
α
1
g
2
)
m
g
r3
3
Tag = mac
r1
(B)
A, B, Tag
-
{
Decommitment phase:
m, r
1
, r
2
, r
3
-
Verify: A
?
= g
r1
1
g
r2
3
B
?
= (g
H(A)
1
g
2
)
m
g
r3
3
Vrfy
r1
(B, Tag)
?
= 1
Figure 4.3: A non-malleable commitment scheme based on the discrete logarithm problem.
a diﬀerent A
′
will be unable to equivocate B
′
(without breaking some computational assumption).
We refer the reader to [40] for further discussion.
In [40], the dependence (upon A) of the public parameters used for commitment B was achieved
via a “selector function,” which results in public parameters whose size is dependent on the length
of the committed message. Here, we exploit algebraic properties to drastically reduce the size of the
public parameters and obtain a more eﬃcient scheme, even in the case of large messages.
4.4.1 Construction Based on the Discrete Logarithm Assumption
The scheme discussed in this section works over any cyclic group G of prime order in which extracting
discrete logarithms is hard but multiplication is easy. For concreteness, we assume an algorithm (
that, on input 1
k
, outputs primes p, q with p = 2q + 1 and [q[ = k; we then take G ⊆ Z
∗
p
to be the
unique subgroup of order q.
Our starting point is the perfect commitment scheme of Pedersen
5
[103], shown in Figure 4.1
(see also the discussion in Section 4.1). The public parameters in our scheme are generated as
follows. First, T T {(1
k
) runs algorithm ((1
k
) to generate primes p, q, thereby deﬁning group G
as discussed above. Next, T T { selects random generators g
1
, g
2
, g
3
of G. Additionally, a random
function H : G → Z
q
is chosen from a family of universal one-way hash functions [98] according to
5
Note that the Pedersen scheme can be made non-interactive by having generators g, h published as part of the
public parameters.
75
algorithm UOWH(1
k
). The output of T T { is (p, q, g
1
, g
2
, g
3
, H).
To commit to a message m ∈ Z
q
(cf. Figure 4.3), the sender ﬁrst chooses random r
1
, r
2
, r
3
∈ Z
q
.
The sender forms the ﬁrst component A by using g
1
, g
3
to “commit” to r
1
. The sender then computes
α = H(A). The second component B is a Pedersen commitment to m with one important diﬀerence:
the ﬁrst generator used for the commitment depends upon α. That is, the sender uses Pedersen
commitment with generators (g
α
1
g
2
) and g
3
. Finally, a Tag of B is computed, using a secure mac
with key r
1
. The security of the commitment scheme is described in the following Theorem:
Theorem 4.2 Assuming (1) the hardness of the discrete logarithm problem for groups G deﬁned
by the output of (, (2) the security of (/, mac, Vrfy) as a message authentication code, and (3)
the security of UOWH as a universal one-way hash family, the protocol of Figure 4.3 is an ε-non-
malleable perfect commitment scheme over ¦Z
q
¦ in the public-parameters model.
Proof It is clear that the protocol is perfectly-hiding since B is uniformly distributed in group G
independently of the message m. We now consider the question of computational binding. Say an
adversary / exists which, when given the public parameters, can output a commitment 'A, B, Tag`
and two legal decommitments 'm, r
1
, r
2
, r
3
` and 'm
′
, r
′
1
, r
′
2
, r
′
3
`, with m = m
′
. Then we can construct
a second adversary /
′
which, given oracle access to /, violates the computational binding of the
standard Pedersen scheme (which holds assuming the discrete logarithm problem is hard [103]). On
input G, g
1
, g
2
, adversary /
′
chooses random s ∈ Z
∗
q
, computes g
3
= g
s
1
, selects a random H, and
runs /(G, g
1
, g
2
, g
2
, H) to generate commitment 'A, B, Tag` and decommitments 'm, r
1
, r
2
, r
3
` and
'm
′
, r
′
1
, r
′
2
, r
′
3
`. Note that / is run on exactly the same distribution of inputs as it would receive in
a real execution of the protocol. Now, /
′
computes α = H(A), and outputs B as its commitment
along with decommitments 'αm + sr
3
, m` and 'αm
′
+ sr
′
3
, m
′
`. If the decommitments produced
by / are legal decommitments to diﬀerent values, then the decommitments output by /
′
are legal
decommitments to diﬀerent values. This proves the computational binding of the original protocol.
The proof of non-malleability is more involved, and we ﬁrst provide some intuition. The simulator
(which will do as well as the adversary without seeing any commitment) ﬁrst generates public
parameters which are distributed identically to the real experiment, but for which the simulator
knows some trapdoor information allowing the simulator to perfectly equivocate its commitment
(cf. Deﬁnition 4.4). The simulator generates a commitment com to a random message, gives this
commitment to the adversary, and receives the commitment com
2
in return. The simulator now tries
to get the adversary to decommit com
2
as some message; this is the message that will be output
by the simulator. To get the adversary to open its commitment, the simulator decommits com to
a random message and gives the decommitment to the adversary, repeating this step (rewinding
76
the adversary each time) a bounded number of times until the adversary opens
6
com
2
. Since the
simulator can perfectly equivocate its commitment, the adversary’s view is equivalent to its view in
the original experiment. Furthermore, we show that the adversary cannot equivocate its commitment
com
2
without contradicting the discrete logarithm assumption. A complete proof follows.
We begin by describing an equivocable commitment generator Equiv that will be used as a
subroutine by our simulator /
′
:
Equiv
1
(1
k
)
p, q ← ((1
k
)
g
1
, g
3
←G; H ← UOWH(1
k
)
r
1
, r
2
, t, u ←Z
q
A := g
r1
1
g
r2
3
; α := H(A)
g
2
:= g
−α
1
g
t
3
σ := 'p, q, g
1
, g
2
, g
3
, H`
B := g
u
3
; Tag := mac
r1
(B)
com := 'A, B, Tag`
s := 'p, q, r
1
, r
2
, t, u`
Output (σ, com, s)
Equiv
2
('p, q, r
1
, r
2
, t, u`, m)
if m / ∈ Z
q
output ⊥
r
3
:= u −tm mod q
dec := 'm, r
1
, r
2
, r
3
`
Output dec
Note that Equiv satisﬁes Deﬁnition 4.4. Furthermore, p, q, g
1
, g
3
can be chosen at random and given
to Equiv; knowledge of log
g1
g
3
is not necessary. This will be crucial for the proof of security.
We now describe the simulator /
′
in more detail. Fix ε, R. Let com = 'A, B, Tag` be the
commitment output by Equiv, and com
2
= 'A
′
, B
′
, Tag
′
` be a commitment output by /
1
. Say event
Collision occurs if H(A) = H(A
′
). The simulator runs Equiv
1
to generate public parameters σ and
a commitment com. The adversary, given these values, outputs some commitment com
2
. If event
Collision occurs, the simulator simply outputs ⊥. Otherwise, the simulator repeatedly “opens” com
a bounded number of times until the adversary de-commits, in some legal way, to a message m
2
.
If this occurs, message m
2
(along with σ and T) is then output by the simulator. If the adversary
never de-commits in a legal way, the simulator simply outputs ⊥. A formal description follows:
/
′
(1
k
)
(σ, com, s) ← Equiv
1
(1
k
)
(T, s
′
) ← /
0
(σ)
(com
2
, s
′′
) ← /
1
(σ, com, s
′
)
if Collision then output (σ, T, ⊥)
Fix random coins ωinΩ
Repeat at most 2ε
−1
ln 2ε
−1
times:
m ← T
dec := Equiv
2
(s, m)
dec
2
:= /
2
(σ, com, dec, s
′′
; ω)
m
2
:= {(σ, com
2
, dec
2
)
if m
2
=⊥ break
output (σ, T, m
2
)
6
If the adversary never opens its commitment, the simulator outputs ⊥.
77
We will show that the diﬀerence Succ
NM
A,R
(k)−
¯
Succ
A
′
,R
(k) (cf. Deﬁnition 4.2) is less than ε+negl(k).
The simulator cannot succeed whenever Collision occurs since it outputs m
2
=⊥ in this case.
This is not a problem, however, since the the probability that both Collision and a success for the
adversary occur must be negligible. To see this, note that event Collision can occur in three ways:
1. 'A, B, Tag` = 'A
′
, B
′
, Tag
′
`. In this case, the adversary cannot succeed by deﬁnition.
2. 'A, B, Tag` = 'A
′
, B
′
, Tag
′
` but A = A
′
. Say the adversary later gives legal decommitment
'm
′
, r
′
1
, r
′
2
, r
′
3
` for com
2
following legal decommitment 'm, r
1
, r
2
, r
3
` for com. There are two
possibilities: either 'r
1
, r
2
` = 'r
′
1
, r
′
2
` or not. If they are not equal, then / has violated the
computational binding property of the Pedersen scheme with generators g
1
, g
3
. If they are equal
then the fact that Vrfy
r1
(B, Tag) = 1 and 'B
′
, Tag
′
` = 'B, Tag` implies that / has violated
the security of the mac (note that r
1
is information-theoretically hidden from the adversary
when com
2
is output). Either of these events can occur with only negligible probability. Thus,
the adversary can give a legal decommitment to com
2
(and hence succeed) with only negligible
probability.
3. A = A
′
but H(A) = H(A
′
). In this case, / has violated the security of the universal one-way
hash family (note that A may be computed by Equiv
1
before H is chosen). This can only occur
with negligible probability.
From now on, assume that event Collision does not occur, since this can only contribute a neg-
ligible quantity to the diﬀerence of interest. Without loss of generality, we further assume that T
output by /
0
(σ) is always a valid distribution over ´
σ
def
= Z
q
since the adversary cannot succeed,
by deﬁnition, when this is not the case. Straightforward manipulation, using the fact that Equiv is
a perfectly equivocable commitment generator and (T T {, o, {) is a perfect commitment scheme
gives
Succ
NM
A,R
(k) =
Pr

(4.5)
+ε/2.
Note that if Pr[γ ← Γ(1
k
) : Good] = 0 we are done, since the above expression is then equal to
ε/2. Assuming that event Good occurs with non-zero probability, we now bound (4.4) and (4.5).
First, notice that expression (4.4) is bounded from above by the probability that m
∗
2
=⊥. However,
deﬁnition of event Good and a straightforward probability calculation show that:
Pr

2
−ε(k)
p
A
(k)
,
But this immediately yields
¯ p
A
′ (k) ≥ p
A
(k) −
ε(k)
p
A
(k)
and therefore p
A
(k) − ¯ p
A
′ (k) ≤ negl(k) which is the desired result.
4.4.2 Construction Based on the RSA Assumption
Here, our starting point is the RSA-based perfect commitment scheme of Okamoto [100]. Let N
be a product of two primes, and let g ∈ Z
∗
N
and e a prime number be given. Then, a commitment
to a message m ∈ Z
e
is generated by choosing a random u ∈ Z
∗
N
and forming the commitment
g
m
u
e
mod N. It is easy to see that this scheme achieves information-theoretic secrecy. We use the
following lemma to show that the scheme is computationally binding under the RSA assumption.
Lemma 4.2 Fix N. For any g ∈ Z
∗
N
and non-zero e and m such that gcd([m[, [e[) = 1, given
u ∈ Z
∗
N
such that g
m
= u
e
mod N, we may eﬃciently compute g
1/e
mod N.
Proof Without loss of generality, we may assume that e, m > 0; if, for example, m < 0 we can
re-write the above equation as (g
−1
)
−m
= u
e
mod N. Since e and m are relatively prime, we may
eﬃciently compute (using the extended Euclidean algorithm) integers a, b such that am + be = 1.
But then:
g
1
= g
am
g
be
= u
ae
g
be
mod N
=

e
.
Since [∆[ < e, [(m
2
− m
∗
2
)[ < e, and e is prime, we have that ∆ (m
2
− m
∗
2
) and e are relatively
prime. Application of Lemma 4.2 shows that we can then compute g
1/e
.
4.5 Extensions
Arbitrarily-long messages. Theorems 4.2 and 4.3 hold even if the message is hashed before
commitment (note that hashing the message before commitment is not known to be secure for an
85
arbitrary non-malleable commitment scheme; in fact, evidence to the contrary is given by the con-
struction of [58]). To see this, note that Equiv can still perfectly equivocate to any (random) message
M by ﬁrst computing m = H(M) and then running the identical Equiv
2
algorithm. The simulator
/
′
is also identical. The hash function H must be collision resistant for the binding property to
hold, but no other assumptions about the hash function are necessary, and the scheme maintains
perfect secrecy.
7
. The present schemes therefore give practical and provably-secure methods for
non-malleable, perfect commitment to arbitrarily long messages.
Reducing the commitment size. Our schemes produce commitments com = (A, B, Tag) of
size (roughly) 2k, where k is the length of a string representing a group element. However, one
can replace this commitment with any string that uniquely binds the sender to com. At least two
modiﬁcations in this vein seem useful:
• Using a collision-resistant hash-function h, we can replace the commitment com with h(com).
The decommitment phase is the same as before. This does not increase the computational
cost of the protocol by much. The resulting commitment size is equal to the output length
of a hash function believed to be collision-resistant. In particular, this allows us to achieve
optimal commitment size ω(log k), assuming an appropriate hash function. Note that hashing
the commitment is not known to give provable security for general non-malleable commitment
schemes, yet it does work (as can be seen by careful examination of the proof) for the particular
constructions given here.
• By adding one more public parameter and making appropriate (small) modiﬁcations, we can
(for example, in Figure 4.3) set the commitment to the product of A, B and Tag (assuming
A is an extended-Pedersen commitment to r
1
, r
2
and Tag is computed as B
r1
g
r2
3
, which is an
information-theoretically secure mac of B). This reduces the commitment length to k.
Unique identiﬁers. As mentioned in [44], in many situations there is a unique identiﬁer associated
with each user and using this can improve the eﬃciency of non-malleable primitives. This is also true
of our perfect commitment schemes. For example, in our discrete-logarithm construction, if each
user in the system has identiﬁer id ∈ Z
q
, we can simplify the scheme by replacing α with id. An
adversary who attempts to generate related commitments must do so with respect to his identiﬁer
id
′
= id. The commitment is now simply B, as the components A and Tag are no longer needed
(their only role in the original protocol was to force an adversary to change α).
7
This can be compared to [58] which requires added complications when using an arbitrary hash function and
achieves only statistical secrecy.
86
Chapter 5
Non-Malleable and Concurrent
(Interactive) Proofs of Knowledge
5.1 Introduction
A proof of knowledge, introduced by Goldwasser, Micali, and Rackoﬀ [70], represents a formalization
of the deceptively simple notion of “proving that you know something” to someone else. More
formally, consider an arbitrary relation R which is computable in polynomial time. Say a prover {
and a veriﬁer 1 have common input x, and { additionally knows a value w such that R(x, w) = 1.
How can { prove to 1 that he indeed knows such a w? Of course, { can simply send w to 1, but,
in many cases, this will reveal more information than { would like. One can try to construct other
interactive protocols for accomplishing this task, but without a precise deﬁnition it is unclear how
to proceed.
Indeed, deﬁning the notion correctly has been diﬃcult [70, 54, 116]. The eﬀort to obtain the
“right” deﬁnition culminated in the work of Bellare and Goldreich [8] which contains the now-
standard deﬁnitional approach. Informally, and omitting many details, the deﬁnition states that
an interactive protocol Π constitutes a proof of knowledge if, for any Turing machine { which
successfully convinces a veriﬁer 1 with “high” probability (where 1 executes Π), the value w may
be extracted from { by an explicitly-given extraction algorithm (the knowledge extractor).
1
We illustrate the above discussion with an example [111]. Let the common input to { and 1
be a ﬁnite, cyclic group G of prime order q, a generator g of G, and a value y ∈ G. Additionally,
assume { has as input a value x ∈ Z
q
such that y = g
x
. Figure 5.1 shows an interactive protocol by
which { can convince 1 that { in fact knows the discrete logarithm x of the common input y. To
begin, { chooses a random value r ∈ Z
q
, computes A = g
r
, and sends A to 1. Then, 1 chooses a
1
Non-interactive proofs of knowledge are possible if the prover and veriﬁer share a common random string [39];
however, since we do not require this notion here we omit further details.
87
Common input: G, g, y
{ (input x ∈ Z
q
)
1
r ←Z
q
A := g
r
A
-
c ←Z
q c

z := cx +r mod q
z
-
Verify: g
z
?
= y
c
A
Figure 5.1: Proof of knowledge of a discrete logarithm.
random challenge c ∈ Z
q
and sends c to {. To respond, { computes z = cx +r mod q and sends z
back to the veriﬁer. 1 checks that g
z
?
= y
c
A. Note that if { is honest (and really knows the correct
value x) veriﬁcation will always succeed since g
z
= g
cx+r
= g
xc
g
r
= y
c
A.
To see intuitively why this is a proof of knowledge, note that there are two possibilities once {
has sent A: either { is able to respond correctly to only one or fewer possible challenges, or { can
respond correctly to two or more challenges. In the former case, {’s probability of “fooling” 1 (who
picks a random challenge) is at most 1/q. In particular, if [q[ = k (where k is a security parameter)
this probability is negligible. On the other hand, if { “knows” correct responses z
1
, z
2
to the two
distinct challenges c
1
, c
2
, this implies that g
z1
= y
c1
A and g
z2
= y
c2
A. But then g
z1−z2
= y
c1−c2
and hence { “could” eﬃciently compute log
g
y = (z
1
−z
2
)/(c
1
−c
2
) mod q himself. In other words,
if { has the ability to convince 1 with non-negligible probability, then {, in some sense, already
must be able to compute x himself. A formal proof that the protocol satisﬁes the formal deﬁnition
of a proof of knowledge [8] is also possible.
Often, a protocol like that of Figure 5.1 is more useful than having { simply send x, since the
protocol is in fact a zero-knowledge protocol as long as 1 is honest. To see this (informally), note
that a random transcript of an execution of the protocol for any y can be eﬃciently simulated even
without knowledge of log
g
y. To simulate, ﬁrst pick random c, z ∈ Z
q
. Then, set A = g
z
/y
c
. The
simulated transcript, which is distributed identically to a real transcript, is (A, c, z). Thus, the
protocol reveals no information about x (beyond what can be computed in polynomial time from y
alone) to an honest veriﬁer 1.
Proofs of knowledge have a wide range of applications. They are crucial for secure two-party and
multi-party computation [66], and have also been used to build interactive commitment protocols
[44, 59] and identiﬁcation schemes [56, 74, 111]. They may also be used to construct encryption
88
{ (input x) ´ 1
r ←Z
q
y := g
x
; A := g
r
y, A
-
r
′
←Z
q
y
′
:= yg
r
′
y
′
, A
-
c ←Z
q
c

c

z := cx +r
z
-
z
′
:= z +cr
′
z
′
-
Verify: g
z
′ ?
= (y
′
)
c
A
Figure 5.2: Man-in-the-middle attack on a proof of knowledge.
schemes with strong security properties [99, 44, 110, 38].
Unfortunately, the standard deﬁnition of a proof of knowledge is not suﬃcient in a network-based
setting such as that considered in this work. More precisely, if an adversary ´ acts as a man-in-
the-middle between { and 1, where { proves knowledge of x to ´ while ´ proves knowledge of x
′
to 1, the standard deﬁnition does not preclude ´ convincing 1 even when ´ does not really know
x
′
. Proofs of knowledge in which such man-in-the-middle attacks are possible are called malleable.
As an example, we demonstrate in Figure 5.2 that the protocol of Figure 5.1 is malleable. In Figure
5.2, we assume that {, ´, and 1 all share the same group G and generator g as common input.
Furthermore, for simplicity, the element y for which knowledge of log
g
y is proved is included with
the ﬁrst message. In the ﬁgure, { proves knowledge of x = log
g
y to ´, while ´successfully proves
knowledge of x
′
= log
g
y
′
to 1; note, however, that ´ does not actually know x
′
!
For many suggested applications of proofs of knowledge, preventing such attacks is essential.
To give just one example, it has been suggested [60, 75, 62] (following [99]) to use interactive
(zero-knowledge) proofs of knowledge to achieve chosen-ciphertext-secure (interactive) public-key
encryption via the following construction: to encrypt a message m using public key pk, the sender
computes C = c
pk
(m; r) for random r, and then executes an interactive proof-of-knowledge (with
the receiver) of m and r. Unfortunately, while this construction is suﬃcient to achieve non-adaptive
chosen-ciphertext security, it does not guarantee adaptive chosen-ciphertext security when the proof
of knowledge is malleable.
The standard deﬁnition of a proof of knowledge is also not suﬃcient when multiple proofs are
executed by multiple provers in a concurrent and asynchronous setting. To see why this is the
89
case, note that extraction of a valid witness from the prover typically requires the knowledge ex-
tractor to rewind the prover. However, if many proofs are being conducted simultaneously in an
arbitrarily-interleaved manner, extracting all witnesses from all provers at the appropriate points
of the execution
2
may require exponential time due to the nested rewindings. A similar problem
arises in the simulation of concurrent zero-knowledge proofs; in that case, however, interaction with
multiple veriﬁers is the source of the problem.
5.1.1 Our Contributions
We focus on proofs of plaintext knowledge (PPKs) in which a sender o proves knowledge of the
contents m of a ciphertext C (a more formal deﬁnition is given below). However, it is clear that our
deﬁnitions and constructions may be extended to proofs of knowledge for more general NP-relations.
We give the ﬁrst deﬁnition of non-malleability for interactive proofs of (plaintext) knowledge, and
show eﬃcient, non-malleable PPKs for the RSA [107], Rabin [104], Paillier [101], and El-Gamal [51]
cryptosystems. We then highlight important applications of these PPKs to (1) chosen-ciphertext-
secure, interactive encryption, (2) password-based authentication and key-exchange in the public-
key model, (3) deniable authentication, and (4) identiﬁcation. We construct eﬃcient and practical
protocols for each of these tasks, improving and extending previous work. Our results include:
• The ﬁrst practical and non-malleable interactive encryption schemes based on the RSA, fac-
toring, or (computational) composite residuosity assumptions.
• The ﬁrst practical protocols for password-based authentication and key-exchange (in the
public-key model) based on the RSA, factoring, or (computational) composite residuosity
assumptions.
• The ﬁrst practical protocols for deniable authentication based on the RSA, factoring, CDH, or
(computational) composite residuosity assumptions. The round-complexities of our protocols
are the same as in the only previous eﬃcient solution based on DDH.
• The ﬁrst 3-round identiﬁcation scheme secure against man-in-the-middle attacks.
Of additional interest, our techniques provide a general methodology for constructing eﬃcient,
non-malleable (zero-knowledge) proofs of knowledge when shared parameters are available. Note that
for the applications listed above, these parameters can simply be included as part of users’ public
keys. Furthermore, this work is the ﬁrst to consider the issues arising in concurrent executions
2
In our applications, a simulator will be required to extract the i
th
witness from prover P
i
as soon as P
i
successfully
completes his proof. Indeed, this is the source of the problem, since extraction of all witnesses at the end of the entire
interaction (i.e., after all provers have completed their proofs) can be done in expected polynomial time.
90
of proofs of knowledge; previous work (e.g., [49]) considered concurrency only in the context of
zero-knowledge.
5.1.2 Previous Work
Proofs of plaintext knowledge are explicitly considered by Aumann and Rabin [1] who provide an
elegant solution for any public-key encryption scheme. Our solutions improve upon theirs in many
respects: (1) by working with speciﬁc, number-theoretic assumptions we vastly improve the eﬃciency
and round-complexity of our schemes; (2) we explicitly consider malleability and ensure that our
solutions are non-malleable; (3) our protocols are secure even against a dishonest veriﬁer, whereas
[1] only considers security against an honest veriﬁer (i.e., the intended recipient); (4) we explicitly
handle concurrency and our protocols remain provably-secure under concurrent composition. More
generally, every NP-relation has a (zero-knowledge) argument of knowledge assuming the existence
of one-way functions [54]; furthermore, once public information is assumed (as we assume here) and
assuming the existence of trapdoor permutations and dense cryptosystems, non-interactive zero-
knowledge (NIZK) proofs of knowledge are also possible [39].
None of the above-mentioned solutions are non-malleable. Dolev, Dwork, and Naor [44] intro-
duce deﬁnitions and constructions for non-malleable, zero-knowledge, interactive proofs. Sahai [110]
subsequently considers the case of non-malleable, non-interactive, zero-knowledge proofs and proofs
of knowledge; he provides deﬁnitions and constructions for the “single-theorem” case and shows ex-
tensions to the case of a bounded-polynomial number of proofs. De Santis, et al. [38] give deﬁnitions
and constructions for robust non-interactive zero-knowledge proofs and proofs of knowledge (which
are, in particular, non-malleable), improving upon previous work. Our deﬁnitions extend those of De
Santis, et al. [38] to the interactive setting; interestingly, ours is the ﬁrst work to explicitly consider
non-malleability for interactive proofs of knowledge.
The above-mentioned works [44, 110, 38] show that, in principal, solutions to our problem ex-
ist. These solutions, however, are impractical. In particular, known non-malleable interactive proofs
[44] require a poly-logarithmic number of rounds, while in the non-interactive setting practical NIZK
proofs — let alone non-malleable ones — are not currently known for number-theoretic problems of
interest (i.e., without reducing the problem to a general NP-complete language). The techniques out-
lined in this paper serve as a general method for achieving practical non-malleable (zero-knowledge)
proofs of knowledge when public parameters are available, a problem which has not been previously
considered.
We discuss previous work relating to non-malleable encryption, password-based key exchange,
deniable authentication, and identiﬁcation in the appropriate sections of this chapter.
91
5.1.3 Outline of the Chapter
We introduce deﬁnitions for proofs of plaintext knowledge (PPKs) and non-malleable PPKs in Sec-
tion 5.2. As mentioned previously, the latter is the ﬁrst deﬁnition of non-malleability for interactive
proofs of knowledge. Sections 5.3.1–5.3.4 describe very eﬃcient constructions of non-malleable PPKs
based on the RSA assumption, the hardness of factoring, the composite residuosity assumption, and
the DDH assumption, respectively. We then consider applications of our non-malleable PPKs.
Sections 5.4.1–5.4.5 introduce deﬁnitions for non-malleable interactive encryption, password-based
authentication and key exchange, deniable (message) authentication, and identiﬁcation schemes se-
cure against man-in-the-middle attacks. These sections also describe practical constructions (based
on our non-malleable PPKs) for these tasks.
5.2 Deﬁnitions and Preliminaries
This section includes deﬁnitions speciﬁcally related to PPKs and non-malleable PPKs only. Other
deﬁnitions appear in the relevant sections of this chapter.
Non-malleable proofs of plaintext knowledge. The deﬁnitions given in this section focus on
proofs of plaintext knowledge, yet they may be easily extended to proofs of knowledge for general NP-
relations. We assume a non-interactive public-key encryption scheme (/, c, T). The encryption of
message m under public key pk using randomness r to give ciphertext C is denoted as C := c
pk
(m; r).
In this case, we say that tuple (m, r) is a witness to the decryption of C under pk. For convenience,
we assume that [pk[ = k, where k is the security parameter. We let the notation 'A(a), B(b)`(c)
be the random variable denoting the output of B following an execution of an interactive protocol
between A (with private input a) and B (with private input b) on joint input c, where A and B have
uniformly-distributed random tapes.
A proof of plaintext knowledge (PPK) allows a sender o to prove knowledge of a witness to
the decryption of some ciphertext C to a receiver {. Both o and { have an additional joint
input σ; in practice, this may be published along with {’s public key pk.
3
To be useful, a PPK
should additionally ensure that no information about m is revealed, either to the receiver (which is
important if the receiver does not have the secret key) or to an eavesdropper. So that no information
about m is revealed, a PPK is required to be zero-knowledge in the following sense: as mentioned
above, our PPKs use parameters σ (known to all parties); these parameters will be generated by an
algorithm ((pk). We require the existence of a simulator o1´ which takes pk as input and outputs
parameters σ whose distribution will be equivalent to the output of ((pk). Furthermore, given any
3
It is important to note that there is no incentive for R to cheat when choosing σ.
92
valid ciphertext C (but no witness to its decryption), o1´ must be able to perfectly simulate a
PPK of C with any (malicious) receiver {
′
using parameters σ.
Our deﬁnitions build on the standard one for proofs of knowledge [8], except that our protocols
are technically arguments of knowledge and we therefore restrict ourselves to consideration of provers
running in probabilistic, polynomial time.
Deﬁnition 5.1 Let Π = ((, o, {) be a tuple of ppt algorithms. We say Π is a proof of plaintext
knowledge (PPK) for encryption scheme (/, c, T) if the following conditions hold:
(Completeness) For all pk output by /(1
k
), all σ output by ((pk), and all C with witness w to
the decryption of C under pk we have 'o(w), {`(pk, σ, C) = 1 (when { outputs 1 we say it accepts).
(Perfect zero-knowledge) There exists a ppt simulator o1´ such that, for all pk output by
/(1
k
), all computationally-unbounded {
′
, and all m, r, the following distributions are equivalent:
¦σ ← ((pk); C := c
pk
(m; r) : 'o(m, r), {
′
`(pk, σ, C)¦
¦(σ, s) ← o1´
1
(pk); C := c
pk
(m; r) : 'o1´
2
(s), {
′
`(pk, σ, C)¦.
(Witness extraction) There exists a function κ : ¦0, 1¦
∗
→ [0, 1], a negligible function ε(), and
an expected-polynomial-time knowledge extractor /c such that, for all ppt algorithms o
′
, with all
but negligible probability over pk output by /(1
k
), σ output by ((pk), and uniformly-distributed r,
machine /c satisﬁes the following:
Denote by p
pk,σ,r
the probability that { accepts when interacting with o
′
(using random
tape r) on joint input pk, σ, C (where C is chosen by o
′
). On input pk, σ, and access to
o
′
r
, the probability that /c outputs a witness to the decryption of C under pk is at least:
p
pk,σ,r
−κ(pk) −ε([pk[).
The zero-knowledge property stated above is quite strong: o1´
2
achieves a perfect simulation
without rewinding {
′
. This deﬁnition is met by our constructions. More generally, one may weaken
the zero-knowledge requirement to allow, for example, computational indistinguishability where
o1´
2
is given oracle access to {
′
(i.e., is allowed to rewind {
′
). While this is an interesting
direction, we do not pursue such a deﬁnition here.
A non-malleable PPK should satisfy the intuition that “anything proven by a man-in-the-middle
adversary ´is known by ´ (unless ´simply copies a proof).” To formalize this idea, we allow ´
to interact with a simulator (whose existence is guaranteed by Deﬁnition 5.1) while simultaneously
interacting with a (real) receiver {. The goal of ´ is to output ciphertexts C, C
′
(which may be
93
chosen adaptively) and then successfully complete a PPK of C
′
to { while the simulator is executing
a PPK of C to ´. The following deﬁnition states (informally) that if { accepts ´’s proof — yet
the transcripts of the two proofs are diﬀerent — then a knowledge extractor /c
∗
can extract a
witness to the decryption of C
′
. The reason we have ´ interact with the simulator instead of the
real sender o is that we must ensure that the knowledge is actually extracted from ´, and not
from the real sender. This deﬁnition is based on the ideas of [38], who deﬁne a similar notion in the
non-interactive setting.
Deﬁnition 5.2 PPK ((, o, {) is non-malleable if there exists a simulator o1´ (satisfying the
relevant portion of Deﬁnition 5.1), a function κ
∗
: ¦0, 1¦
∗
→ [0, 1], a negligible function ε
∗
(), and an
expected-polynomial-time knowledge extractor /c
∗
such that, for all ppt algorithms ´, with all but
negligible probability over pk output by /(1
k
), σ, s output by o1´
1
(pk), and uniformly-distributed
r, r
′
, machine /c
∗
satisﬁes the following:
Assume ´ (using random tape r
′
) acts as a receiver with o1´
2
(s; r) on joint input
pk, σ, C and simultaneously as a sender with { on joint input pk, σ, C
′
(where C is a
valid ciphertext and C, C
′
are adaptively chosen by ´). Let the transcripts of these two
interactions be π and π
′
. Denote by p
∗
the probability (over the random tape of {) that
{ accepts in the above interaction and π = π
′
. On input pk, σ, s, r, and access to ´
r
′ ,
the probability that /c
∗
outputs a witness to the decryption of C
′
under pk is at least:
p
∗
−κ
∗
(pk) −ε
∗
([pk[).
We note that our deﬁnitions of zero-knowledge (in Deﬁnition 5.1) and non-malleability (in Def-
inition 5.2) both consider the single-theorem case. The deﬁnitions may be modiﬁed for the multi-
theorem case; however, the present deﬁnitions suﬃce for our intended applications.
Σ-protocols. Since we use Σ-protocols [31] in an essential way as part of our constructions, we
brieﬂy review their deﬁnition here. A Σ-protocol is a pair of ppt algorithms ({, 1) which deﬁnes a
three-move interactive protocol between a prover { and a veriﬁer 1, where the prover sends the ﬁrst
message. Furthermore, the message q sent by 1 is a random challenge (without loss of generality,
we may assume it is the contents of 1’s random tape). Let R be a binary relation computable
in polynomial-time. Deﬁne L
R
as the set of all y such that there exists an x with R(y, x) = 1.
The string x is called a witness for y. The common input to { and 1 will be y, while x is known
only to {. Let (A, q, z) be a transcript of the conversation between { and 1. Upon completion
of the protocol, the veriﬁer outputs the single bit ϕ(y, A, q, z), where ϕ() is an eﬃcient, publicly-
computable predicate with 1 denoting acceptance and 0 denoting rejection. If ϕ(y, A, q, z) = 1, we
say that (A, q, z) is an accepting conversation for y.
94
All Σ-protocols we consider satisfy special soundness and special honest-veriﬁer zero-knowledge
(special-HVZK). For a particular y, let (A, q, z) and (A, q
′
, z
′
) denote two accepting conversations
with q = q
′
. Special soundness implies that y ∈ L
R
and furthermore, that on input y and these
two conversations, one can eﬃciently compute an x such that R(y, x) = 1. Special-HVZK means
that there exists a ppt simulator o that, on input y ∈ L
R
and a randomly-chosen q, can generate
a conversation (A, q, z) which is identically-distributed to a real conversation between { and 1 in
which q is the challenge sent by 1. Note that the protocol of Figure 5.1 is a Σ-protocol satisfying
the above requirements.
A note on complexity assumptions. Our proofs of security require hardness assumptions with
respect to adversaries permitted to run in expected polynomial time. For example, we assume that the
RSA function cannot be inverted with more than negligible probability by any expected-polynomial-
time algorithm. The reason for this is our reliance on constant-round proofs of knowledge, for which
only expected-polynomial-time knowledge extractors are currently known. Security deﬁnitions of
protocols, however, are stated with respect to ppt adversaries.
5.3 Non-Malleable Proofs of Plaintext Knowledge
Our constructions follow a common paradigm. Recall the parameter σ which is shared by the sender
and receiver and which is used as a common input during execution of the PPK. Embedded in σ
will be a particular value y for which the simulator knows a witness x such that R(y, x) = 1. A PPK
for ciphertext C will be a witness indistinguishable proof of knowledge of either a witness to the
decryption of C or a witness for y, using the generic techniques for constructing such proofs [35].
Note that soundness of the protocol is not aﬀected since a ppt adversary cannot derive a witness
for y, while ZK simulation is easy (since the simulator knows the witness x).
As stated, this simple approach does not suﬃce to achieve non-malleability. To see why, consider
a simulator interacting with man-in-the-middle ´ while ´ simultaneously interacts with veriﬁer
1. Since the simulator must simulate a proof of “w or x” (where w is the witness to the decryption
of the ciphertext C chosen by the adversary), the simulator must know x. However, if we use the
knowledge extractor to extract from ´, who is proving “w
′
or x” (where w
′
is the witness to the
decryption of C
′
), there is nothing which precludes extracting x! Note that without initial knowledge
of x the simulator cannot properly perform the simulation; yet, if the simulator initially knows x,
there is no contradiction in extracting this value from ´. Thus, a more careful approach is needed.
To overcome this obstacle, we borrow a technique from Chapter 4. Namely, the value y will
depend on a parameter α which ´ cannot re-use; thus, the simulator proves knowledge of “w or
x
α
” while ´ is forced to prove knowledge of “w
′
or x
α
′ ”, for some α
′
= α. This will be secure as
95
long as the following conditions hold: (1) it is possible for the simulator to know the witness x
α
; yet
(2) learning x
α
′ for any α
′
= α results in a contradiction; furthermore, (3) ´ cannot duplicate the
value α used by the simulator. Details follow in the remainder of this section.
5.3.1 Construction for the RSA Cryptosystem
We brieﬂy review the RSA cryptosystem, extended to allow encryption of ℓ-bit messages using the
techniques of Blum and Goldwasser [21]. The public key N is chosen as a product of two random
k/2-bit primes (where k is the security parameter), and e is a prime number such that [e[ = O(k).
4
Let hc() be a hard-core bit [64] for the RSA permutation (so that, given r
e
, hc(r) is computationally
indistinguishable from random; note that hc() may depend on information included with the public
parameters), and deﬁne hc
∗
(r)
def
= hc(r
e
ℓ−1
)◦ ◦hc(r
e
)◦hc(r). Encryption of ℓ-bit message m is done
by choosing a random element r ∈ Z
∗
N
, computing C = r
e
ℓ
mod N, and sending 'C, c
def
= hc
∗
(r)⊕m`.
It is easily shown that this scheme is semantically secure under the RSA assumption.
Our protocol uses a Σ-protocol for proving knowledge of e
ℓ
-th roots which extends a previously-
given Σ-protocol for proving knowledge of e-th roots [74]. To prove knowledge of r = C
1/e
ℓ
, the
prover chooses a random element r
1
∈ Z
∗
N
and sends A = r
e
ℓ
1
to the veriﬁer. The veriﬁer replies
with a challenge q selected randomly from Z
e
. The prover responds with R = r
q
r
1
and the receiver
veriﬁes that R
e
ℓ ?
= C
q
A. To see that special soundness holds, consider two accepting conversations
(A, q, R) and (A, q
′
, R
′
). Since R
e
ℓ
= C
q
A and (R
′
)
e
ℓ
= C
q
′
A we have (R/R
′
)
e
ℓ
= C
q−q
′
. Noting
that [q − q
′
[ is relatively prime to e
ℓ
, Lemma 4.2 shows that the desired witness C
1/e
ℓ
may be
eﬃciently computed. Special-HVZK is demonstrated by the simulator which, on input C and a
“target” challenge q, generates A = R
e
ℓ
/C
q
for random R ∈ Z
∗
N
and outputs transcript (A, q, R).
We now describe the non-malleable PPK in detail (cf. Figure 5.3). The public parameters
σ (which may be included with the public key) are generated by selecting two random elements
g, h ∈ Z
∗
N
. Additionally, a function H : ¦0, 1¦
∗
→ Z
e
from a family of universal one-way hash
functions is chosen at random. Once σ is established, a PPK for ciphertext 'C, c` proceeds as follows:
ﬁrst, a key-generation algorithm for a one-time signature scheme is run to yield veriﬁcation key VK
and signing key SK, and α = H(VK) is computed. The PPK will be a witness indistinguishable
proof of knowledge of either r = C
1/e
ℓ
or x
α
def
= (g
α
h)
1/e
, following the paradigm of [35]. The
sender chooses random elements r
1
, R
2
∈ Z
∗
N
along with a random element q
2
∈ Z
e
. The sender
then executes a real proof of knowledge of C
1/e
ℓ
(using the known witness) and a simulated proof
of knowledge of (g
α
h)
1/e
for challenge q
2
(using the simulator guaranteed by the special-HVZK
property of the Σ-protocol). In more detail, element A
1
is computed as r
e
ℓ
1
while A
2
is computed
4
Below, we mention a modiﬁcation of the protocol for the case of small e (e.g., e = 3).
96
Public key: N; prime e
σ : g, h ∈ Z
∗
N
; H : {0, 1}
∗
→ Ze
o (input m ∈ ¦0, 1¦
ℓ
) {
(VK, SK) ← SigGen(1
k
)
r, r
1
, R
2
←Z
∗
N
; q
2
←Z
e
C := r
e
ℓ
; c := hc
∗
(r) ⊕m
α := H(VK)
A
1
:= r
e
ℓ
1
A
2
:= R
e
2
/ (g
α
h)
q2
VK, C, c, A
1
, A
2
-
q ←Z
e
q

(q−q1 mod e)
A
2
Vrfy
VK
(transcript, s)
?
= 1
Figure 5.3: Non-malleable PPK for the RSA cryptosystem.
as R
e
2
/(g
α
h)
q2
. These values are sent (along with VK, C, c) as the ﬁrst message of the PPK. The
receiver chooses challenge q ∈ Z
e
as before. The sender sets q
1
= q − q
2
mod e and answers with
R
1
= r
q1
r
1
(completing the “real” proof of knowledge with challenge q
1
) and R
2
(completing the
“simulated” proof of knowledge with challenge q
2
). The values q
1
, R
1
, R
2
are sent to the receiver. To
complete the proof, the sender signs a transcript of the entire execution of the PPK (including C, c
but not including VK itself) using SK and sends the signature to the receiver. The receiver veriﬁes
the correctness of the proofs by checking that R
e
ℓ
1
?
= C
q1
A
1
and R
e
2
?
= (g
α
h)
(q−q1 mod e)
A
2
. Finally,
the receiver veriﬁes the correctness of the signature on the transcript.
Theorem 5.1 Assuming the hardness of the RSA problem for expected-polynomial-time algorithms,
the protocol of Figure 5.3 is a PPK (with κ(pk) = 1/e) for the RSA encryption scheme outlined
above.
Proof We show that the given protocol satisﬁes Deﬁnition 5.1. Completeness is trivial. Sim-
ulatability (zero-knowledge) is achieved by making sure the simulator knows appropriate secret
information about σ. For example, o1´
1
(N, e) may choose random elements x, y and a hash func-
tion H and output σ = 'x
e
, y
e
, H`. Note that σ has the correct distribution. When o1´
2
is
97
requested to run the PPK on any ciphertext, it can do so easily (without rewinding the potentially
dishonest receiver) because it knows (g
α
h)
1/e
for any α. Witness indistinguishability of the proof
(cf. [35]) implies that the simulated transcript is distributed identically to a real transcript.
The witness extraction property follows from the stronger result proved in Theorem 5.2.
Theorem 5.2 Assuming (1) the hardness of the RSA problem for expected-polynomial-time algo-
rithms, (2) the security of (SigGen, Sign, Vrfy) as a one-time signature scheme, and (3) the security
of UOWH as a universal one-way hash family, the protocol of Figure 5.3 is a non-malleable PPK
(with κ
∗
(pk) = 1/e) for the RSA encryption scheme outlined above.
Proof We use the following simulator (which is diﬀerent from the one given in the proof of
Theorem 5.1): o1´
1
(N, e) chooses random hash function H, runs SigGen(1
k
) to generate (VK, SK),
and computes α = H(VK). Random elements g, x ∈ Z
∗
N
are chosen, and h is set equal to g
−α
x
e
.
Finally, σ = 'g, h, H` is output along with state information state = 'VK, SK, x`. Note that σ
output by o1´
1
has the correct distribution. Furthermore, given state, simulation of a single PPK
by o1´
2
for any ciphertext is easy: simply use veriﬁcation key VK and then (g
α
h)
1/e
is known (cf.
the proof of Theorem 5.1). Note that the simulation is perfect due to the witness indistinguishability
of the proof.
Fix pk, σ, state, and randomness r for o1´
2
(recall that ciphertext 'C, c`, for which o1´
2
will be required to prove a witness, is chosen adaptively by ´). We are given adversary ´ using
(unknown) random tape r
′
who interacts with both o1´
2
(state; r) and honest receiver {. Once
the challenge q
′
of { is ﬁxed, the entire interaction is completely determined; thus, we may deﬁne
π(q
′
) as the transcript of the conversation between o1´
2
(state; r) and ´
r
′ when q
′
is the challenge
of {; analogously, we deﬁne π
′
(q
′
) as the transcript of the conversation between ´
r
′ and { when
q
′
is the challenge of {. If certain messages have not been sent (e.g., ´ never sends a challenge q
to o1´
2
) we simply set those messages to ⊥.
The knowledge extractor /c
∗
is given pk, σ, state, r, and access to ´
r
′ . When we say that
/c
∗
runs ´
r
′ with challenge q we mean that /c
∗
interacts with ´
r
′ by running algorithm
o1´
2
(state; r) and sending challenge q for {. We stress that interleaving of messages (i.e., schedul-
ing of messages to/from { and o1´
2
) is completely determined by ´
r
′ . /c
∗
operates as follows:
First, /c
∗
picks a random value q
′
1
∈ Z
e
and runs ´
r
′ with challenge q
′
1
(cf. Figure 5.4). If π
′
(q
′
1
)
is not accepting, or if π
′
(q
′
1
) = π(q
′
1
), stop and output ⊥. Otherwise, run the following:
98
o1´
2
(state; r) ´ {
VK, C, c, A
1
, A
2
-
VK
′
, C
′
, c
′
, A
′
1
, A
′
2
-
q
′
1

poly(k)
p
∗
= poly(k), where poly(k) is an upper bound on the running time of
´. On the other hand, if p
∗
≤ 1/e, the number of iterations of the loop above is at most e, yet the
probability of executing the loop is at most 1/e. Thus, the expected running time of Ext in this case
is at most
1
e
e poly(k) = poly(k).
Let Good be the event that /c
∗
does not output ⊥. In this case, let the transcripts be as
99
indicated in Figure 5.4. Note that the probability of event Good is exactly p
∗
when p
∗
> 1/e and 0
otherwise. In either case, we have Pr[Good] ≥ p
∗
−1/e.
Assuming event Good occurs, π
′
(q
′
1
) and π
′
(q
′
2
) are accepting transcripts with q
′
1
= q
′
2
and
therefore we must have either q
′
1
1
= q
′
1
2
or q
′
1
− q
′
1
1
= q
′
2
− q
′
1
2
mod e (or possibly both). In case
q
′
1
− q
′
1
1
= q
′
2
− q
′
1
2
mod e (denote this event by Real) and hence q
′
1
1
= q
′
1
2
, we have the two
equations:

c−c1 mod q
A
2
V
VK
(transcript, s)
?
= 1
Figure 5.7: Non-malleable PPK for the El Gamal cryptosystem.
VK and SK for a one-time signature scheme and computing α = H(VK). The PPK will be a witness
indistinguishable proof of knowledge of y ∈ Z
q
such that either g
y
0
= C
0
or g
y
0
= g
α
1
h.
Theorem 5.5 Assuming (1) the hardness of the discrete logarithm problem in G for expected-
polynomial-time algorithms, (2) the security of (SigGen, Sign, Vrfy) as a one-time signature scheme,
and (3) the security of UOWH as a universal one-way hash family, the protocol of Figure 5.7 is a
non-malleable PPK (with κ(pk) = 1/q) for the El Gamal encryption scheme.
Proof The proof that the scheme is a PPK is similar to the proof of Theorem 5.1, and is omitted.
Proof of non-malleability follows the outline of the proof of Theorem 5.2, so we only sketch the key
diﬀerences here.
Our simulator is as follows: o1´
1
(p, q, g
0
, g
1
) chooses random hash function H, runs the key-
generation algorithm for the one-time signature scheme to obtain (VK, SK), and computes α =
H(VK). Random x ∈ Z
q
is chosen, and h is set equal to g
−α
1
g
x
0
. Finally, σ = 'h, H` is output along
with state information state = 'VK, SK, x`. Note that σ output by o1´
1
is identically distributed
to σ in a real execution. Furthermore, given state, simulation of a single PPK by o1´
2
for any
valid ciphertext is easy since log
g0
(g
α
1
h) is known.
110
Fix pk, σ, state, and randomness r for o1´
2
. Recall that ciphertext C
0
, C
1
for which o1´
2
will be required to prove a witness is chosen adaptively by ´. We are given adversary ´ using
(unknown) random tape r
′
who interacts with both o1´
2
(state; r) and honest receiver {. For
any query c sent by {, deﬁne π(c) and π
′
(c) as in the proof of Theorem 5.2. As in the proof of
Theorem 5.2, we may deﬁne a knowledge extractor /c
∗
which runs in expected polynomial time
(the proof is similar to that given previously) and outputs either ⊥ or 'π(c
′
1
), π
′
(c
′
1
), π
′
(c
′
2
)`, where
π(c
′
1
) = π
′
(c
′
1
) and π
′
(c
′
1
), π
′
(c
′
2
) are accepting transcripts with c
′
1
= c
′
2
. Let Good be the event
that /c
∗
does not output ⊥. By a similar proof as above, we have Pr[Good] ≥ p
∗
−1/q.
Assuming event Good occurs, π
′
(c
′
1
) and π
′
(c
′
2
) are accepting and therefore we must have either
c
′
1
1
= c
′
1
2
or c
′
1
− c
′
1
1
= c
′
2
− c
′
1
2
mod q (or possibly both). In case c
′
1
−c
′
1
1
= c
′
2
−c
′
1
2
mod q and
hence c
′
1
1
= c
′
1
2
, we show how to compute a value r such that r = log
g0
C
′
0
. From the accepting
transcripts, we have the two equations:
g
z
′
1
1
0
= (C
′
0
)
c
′
1
1
A
′
1
g
z
′
1
2
0
= (C
′
0
)
c
′
1
2
A
′
1
.
Together, these imply g
∆z
0
= (C
′
0
)
∆c
, where ∆
z
def
= z
′
1
1
−z
′
1
2
and ∆
c
def
= c
′
1
1
−c
′
2
2
. Hence, log
g0
C
′
0
=
∆
z
/∆
c
and we are done.
On the other hand, if c
′
1
− c
′
1
1
= c
′
2
− c
′
1
2
mod q, we denote this event by Fake. To complete
the proof, we show that Pr[Fake] (where the probability is over the random tape ω of /c
∗
) is less
than some negligible function with all but negligible probability over choice of pk, σ, state, r, r
′
. We
establish this using Claims 5.1 and 5.2 from the proof of Theorem 5.2 along with the following:
Claim 5.6 Pr
pk,σ,state,r,r
′
,ω
[H(VK) = H(VK
′
) ∧ Fake] is negligible.
Note that algorithm /c
∗
does not require any information about g
0
or g
1
in order to run; in particu-
lar, elements g
0
, g
1
may be selected at random. Thus, we can consider the expected polynomial-time
algorithm /c
′
which takes input modulus p, q, g
0
, g
1
and otherwise runs identically to /c
∗
. Clearly,
the probability that both Fake and H(VK) = H(VK
′
) occur remains unchanged.
We show that log
g0
g
1
may be computed when both Fake and H(VK) = H(VK
′
) occur, contradict-
ing the discrete logarithm assumption. Let α = H(VK) and α
′
= H(VK
′
); deﬁne ∆
α
def
= α
′
−α = 0.
From the two accepting transcripts, we have the two equations:
g
z
′
2
1
0
= (g
α
′
1
h)
c
′1
−c
′
1
1
A
′
2
g
z
′
2
2
0
= (g
α
′
1
h)
c
′2
−c
′
1
2
A
′
2
,
which yield:
g
∆z
0
= (g
α
′
1
h)
∆c
111
= (g
∆α
1
g
x
0
)
∆c
= g
∆α∆c
1
g
x∆c
0
,
where ∆
z
def
= z
′
2
1
− z
′
2
2
and ∆
c
def
= c
′
1
− c
′
1
1
− c
′
2
+ c
′
1
2
= 0 mod q. From this, we may immediately
conclude that log
g0
g
1
=
∆z−x∆c
∆α∆c
.
The remainder of the proof exactly follows that of Theorem 5.2.
5.4 Applications
In this section, we discuss applications of non-malleable PPKs to the construction of (1) chosen-
ciphertext-secure, interactive encryption protocols, (2) password-based authentication and key ex-
change protocols in the public-key model, (3) strong deniable-authentication protocols, and (4)
identiﬁcation protocols secure against man-in-the-middle attacks. In each case, we show how the
protocols of the previous section may be used for the intended application.
Concurrent composition. In our intended applications, the man-in-the-middle adversary may
conduct multiple PPKs concurrently and witness extraction will be required from each such exe-
cution; furthermore, extraction of this witness is typically required as soon as the relevant proof is
completed. If arbitrary interleaving of the proofs is allowed, extracting all witnesses may require
exponential time due to the nested rewinding of the prover (a similar problem is encountered in
simulation of concurrent zero-knowledge proofs [49]). To avoid this problem, we introduce timing
constraints [49] in our protocols. These are explained in detail in the relevant sections.
5.4.1 Chosen-Ciphertext-Secure Interactive Encryption
Previous work. Deﬁnitions for chosen-ciphertext-secure public-key encryption were given by Naor
and Yung [99] and and Rackoﬀ and Simon [105]. Naor and Yung also give a construction achieving
non-adaptive chosen-ciphertext security [99]. The notion of non-malleable public-key encryption
was put forth by Dolev, Dwork, and Naor [44]. The ﬁrst construction of a non-malleable (and
hence chosen-ciphertext-secure [16]) public-key encryption
5
scheme was given in [44], and improved
constructions appear in [110, 38]. These constructions, however, are based on general assumptions
and are therefore impractical. Eﬃcient non-malleable encryption schemes are known in the random
oracle model (e.g., OAEP [14]); we work in the standard model only. Prior to this work, the only
eﬃcient non-malleable encryption scheme in the standard model was [36], whose security is based
on the DDH assumption. Subsequent to the present work, Cramer and Shoup [37] have proposed
non-malleable encryption schemes based on alternate assumptions; yet, it is important to note that
5
Unless stated otherwise, “encryption” refers to non-interactive encryption.
112
the security of these schemes is based on the hardness of decisional problems, whereas we present
schemes whose security may be based on the hardness of computational problems.
Chosen-ciphertext security for interactive public-key encryption has been considered previously
[60, 75, 62], although formal deﬁnitions do not appear until [44]. Using an interactive PPK to
achieve chosen-ciphertext-secure, interactive public-key encryption has been previously proposed
[60, 75, 62]; however, such an approach is not secure against adaptive chosen-ciphertext attacks
unless a non-malleable PPK is used. A practical, non-malleable interactive public-key encryption
scheme (which does not use proofs of knowledge) is given by [44]; however, this protocol requires
a signature from the receiver, making it unsuitable for use in a deniable authentication protocol
(see below). Moreover, this protocol [44] requires the receiver — for each encrypted message —
to (1) compute an existentially unforgeable signature and (2) run the key generation procedure for
a public-key encryption scheme (often the most computationally intensive step). Our protocols,
optimized for particular number-theoretic assumptions, are more eﬃcient.
Deﬁnitions. A number of deﬁnitional approaches to chosen-ciphertext security in the interactive
setting are possible. For example, the notion of non-malleability [44] may be extended for the case
of interactive encryption. An oracle-based deﬁnition is also possible, and we sketch such a deﬁnition
here.
6
We have a sender, a receiver (where the receiver has published public-key pk), and a man-in-the-
middle adversary ´ who controls all communication between them (cf. Chapter 2). To model this,
we deﬁne an encryption oracle and a decryption oracle to which ´ is given access. The encryption
oracle c
b,pk
plays the role of the sender. The adversary may interact with this oracle multiple times
at various points during its execution, and may interleave requests to this oracle with requests to
the decryption oracle in an arbitrary manner. At the outset of protocol execution, the encryption
oracle picks a bit b at random. An instance of the adversary’s interaction with the oracle proceeds as
follows: ﬁrst, the adversary chooses two messages m
0
, m
1
and sends these to c
b,pk
. The encryption
oracle then executes the encryption protocol for message m
b
. The adversary, however, need not act
as an honest receiver. Since the adversary may have multiple concurrent interactions with c
b,pk
,
the oracle must maintain state between the adversary’s oracle calls. Formally, each instance of the
encryption oracle is associated with a unique label; furthermore, each message the adversary sends
to the oracle must include a label indicating to which encryption-instance the message corresponds.
When c
b,pk
sends the ﬁnal message for a given instance of its execution, we say that instance is
completed.
6
To obtain an equivalent deﬁnition using the language of non-malleability, we would need to deﬁne a notion of
non-malleability with respect to vectors of ciphertexts. Such a deﬁnition becomes cumbersome to work with in the
interactive setting.
113
The decryption oracle T
sk
plays the role of a receiver. Since the adversary may perform multiple
concurrent executions of the protocol with the oracle, T
sk
must also record state between oracle calls,
and each message sent by the adversary must also include a label indicating to which decryption-
instance the message corresponds (as above). The adversary need not act as an honest sender. Each
time a given decryption-instance is completed, the decryption oracle computes the decryption (using
the secret key) and sends the resulting message (or ⊥, if the transcript was invalid) to the adversary.
The adversary succeeds if it can guess the bit b. Clearly, some limitations must be placed on
the adversary’s access to the decryption oracle or else the adversary may simply forward messages
between c
b,pk
and T
sk
and therefore trivially determine b. At any point during the adversary’s
execution, the set of transcripts of completed encryption-instances of c
b,pk
is well deﬁned. Addi-
tionally, when a decryption-instance of T
sk
is completed, the transcript of that interaction is well
deﬁned. Upon completing a decryption-instance, let ¦π
1
, . . . , π
ℓ
¦ denote the transcripts (not includ-
ing instance labels) of all completed encryption-instances. We allow the adversary to receive the
decryption corresponding to a decryption-instance with transcript π
′
only if π
′
= π
i
for 1 ≤ i ≤ ℓ.
Deﬁnition 5.3 Let Π = (/, c, T) be an interactive, public-key encryption scheme. We say that Π
is CCA2-secure if, for any ppt adversary A, the following is negligible (in k):

Pr

(sk, pk) ← /(1
k
); b ← ¦0, 1¦ : A
E
b,pk
,D
sk
(1
k
, pk) = b

−1/2

,
where A’s access to T
sk
is restricted as discussed above.
A straightforward hybrid argument shows that it is suﬃcient to consider adversaries which are
allowed only a single access to the encryption oracle. We consider this type of adversary in what
follows.
Constructions. The protocols of Figures 5.3, 5.5–5.7 are in fact chosen-ciphertext-secure interactive
encryption schemes (under the relevant assumptions) when the adversary is given sequential access
to the decryption oracle. More precisely, given a semantically-secure encryption scheme (/, c, T)
(which may be the RSA, Rabin, Paillier, or El Gamal scheme) and non-malleable PPK ((, o, {)
for this encryption scheme, the interactive, chosen-ciphertext-secure encryption scheme (/
′
, c
′
, T
′
)
is deﬁned as follows:
• /
′
(1
k
) runs /(1
k
) to generate pk, sk. Additionally, ((pk) is run to give σ. The public key pk
′
is 'pk, σ` and the secret key is sk.
• To encrypt message m under public key pk
′
= 'pk, σ`, the sender computes C ← c
pk
(m) and
then executes algorithm o for C using parameters σ (i.e., the sender proves knowledge of a
witness to the decryption of C).
114
• To decrypt, the receiver uses { to determine whether to accept or reject the proof of ciphertext
C. If the receiver accepts (we say the proof succeeds), the receiver outputs T
sk
(C). If the
proof is rejected (we say the proof fails), the receiver outputs ⊥.
As mentioned previously, timing constraints are needed to ensure security against an adversary
who is given concurrent access to the decryption oracle. In this case, we require that o respond to the
challenge (i.e., send the third message of the protocol) within time α from when the second message
of the protocol is sent. If o does not respond in this time, the proof is rejected. Additionally, the
protocols are modiﬁed so that a fourth message is sent from the receiver to the sender; this message
is simply an acknowledgment message which is ack if the sender’s proof was veriﬁed to be correct and
⊥ otherwise. Furthermore, { delays the sending of this message until at least time β has elapsed
from when the second message of the protocol was sent (with β > α).
We stress that, when concurrent access to the decryption oracle is allowed, the decryption oracle
enforces the above timing constraints by (1) rejecting any proofs for which more than time α has
elapsed between sending the second message and receiving the third message, and (2) the decrypted
ciphertext is not given to the adversary until after the acknowledgment message is sent (in particular,
until time β has elapsed since sending the second message).
Theorem 5.6 Assuming (1) the hardness of the RSA problem for expected-polynomial-time algo-
rithms, (2) the security of (SigGen, Sign, Vrfy) as a one-time signature scheme, and (3) the security
of UOWH as a universal one-way hash family, the protocol of Figure 5.3 (with [e[ = Θ(k)) is an
interactive encryption scheme secure against sequential chosen-ciphertext attacks. If timing con-
straints are enforced as outlined above, the protocol is secure against concurrent chosen-ciphertext
attacks.
Proof We prove security for the more challenging case of concurrent access to the decryption
oracle. The protocol Π of Figure 5.3 is a PPK for encryption scheme (/, c, T) in which /(1
k
) outputs
as the public key a k-bit modulus N and a k-bit prime e. Encryption of ℓ-bit message m is done by
choosing random r ∈ Z
∗
N
and sending
¯
C = 'r
e
ℓ
, hc
∗
(r) ⊕ m`, where hc
∗
() is a hard-core function
for the RSA permutation. Assuming the hardness of the RSA problem for expected-polynomial-
time algorithms, (/, c, T) is semantically secure against expected-polynomial-time adversaries. We
transform any ppt adversary / mounting a CCA2 attack against Π into an expected-polynomial-
time adversary /
′
attacking the semantic security of (/, c, T). Furthermore, we show that the
advantage of /
′
is not negligible if the advantage of / is not negligible. This will immediately imply
CCA2 security of Π.
Let t(k), which is polynomial in k, be a bound on the number of times / accesses the decryption
115
oracle when run on security parameter 1
k
; without loss of generality, we assume that t(k) ≥ k (for
convenience, in the remainder of the proof we suppress the dependence on k and simply write t).
In the real experiment Expt
0
, the ﬁnal output b
′
of / is completely determined by pk
′
= 'pk, σ`,
random coins r
′
for /, the vector of challenges q = q
1
, . . . , q
t
used during the t instances / interacts
with the decryption oracle, and the randomness used by the encryption oracle (this includes the bit
b, the randomness ω used for the encryption, and the randomness used for execution of the PPK).
Let Succ denote the event that b
′
= b, and let Pr
0
[Succ] denote the probability of this event in the
real experiment.
We modify the real experiment, giving Expt
1
, as follows. Key generation is done by running
/(1
k
) to generate pk, sk. Additionally, o1´
1
(pk) (cf. the proof of Theorem 5.2) is run to generate
parameters σ and state (in the real experiment σ was generated by ((pk)). The public key pk
′
is
'pk, σ` and the secret key is sk. The adversary’s calls to the decryption oracle are handled as in
Expt
0
(in particular, any ciphertext
¯
C may be decrypted since sk is known), but the adversary’s
encryption oracle call will be handled diﬀerently. When / calls the encryption oracle on messages
m
0
, m
1
, we pick b randomly, compute
¯
C
∗
= c
pk
(m
b
; ω) for random ω, and simulate the PPK for
¯
C
∗
using algorithm o1´
2
(state; r) with randomly-chosen r. Now, the ﬁnal output b
′
of / is completely
determined by pk
′
= 'pk, σ`, random coins r
′
for /, the vector of challenges q used by the decryption
oracle, the values b and ω used in computing
¯
C
∗
, and the values state and r used by o1´
2
in
simulating the encryption oracle. Since o1´ yields a perfect simulation of a real execution of the
PPK, we have Pr
1
[Succ] = Pr
0
[Succ], where the ﬁrst probability refers to the probability of an event
in Expt
1
.
We now describe our adversary /
′
attacking the semantic security of (/, c, T). Given the
public key pk, adversary /
′
runs o1´
1
(pk) to generate parameters σ and state. /
′
then ﬁxes the
randomness r
′
of /, and runs / on input pk
′
= 'pk, σ`. Simulation of the encryption oracle for
/ is done as follows: when / submits two messages m
0
, m
1
, adversary /
′
simply forwards these
to its encryption oracle and receives in return a ciphertext
¯
C
∗
(note that the encryption oracle
thus implicitly deﬁnes values b
∗
and ω
∗
). Then, /
′
simulates the PPK for
¯
C
∗
using algorithm
o1´
2
(state; r) for randomly-chosen coins r. Simulation of the decryption oracle for / is done by
choosing a random vector of queries q
∗
and attempting to extract the relevant witnesses (in expected
polynomial time) from the PPKs given by /. Details of the simulation are described below. In case
the simulation is successful, the ﬁnal output b
′
is just the ﬁnal output of /; if the simulation is
not successful, the ﬁnal output b
′
is a randomly-chosen bit. As we show below, the simulation will
succeed with suﬃciently high probability such that if the advantage of / (in attacking the CCA2
security of Π) is not negligible then the advantage of /
′
(in attacking the semantic security of
116
(/, c, T) is not negligible as well. This will complete the proof.
It remains to show how to simulate the decryption oracle. Our proof requires techniques used in
an analysis of concurrent composition of zero-knowledge proofs [49]. We assume that / controls the
scheduling of all messages to and from all the oracles; so, for example, the decryption oracle does
not send its next message until / requests it. Deﬁne the i
th
instance of the decryption oracle as the
i
th
time / requests the second message (i.e., the challenge) of the PPK be sent by the decryption
oracle. In any transcript of the execution of /, we let C
i
denote the ciphertext sent by / in the
i
th
instance of the decryption oracle. For any instance of the decryption oracle, we say the instance
succeeds if (1) an honest receiver would accept the instance, (2) the transcript of the instance is
diﬀerent from the transcript (if it yet exists) of the interaction of / with the encryption oracle, and
(3) the timing constraints are satisﬁed for that instance. Otherwise, we say the instance fails.
Recall that the simulator has values 'pk, σ, r
′
, q
∗
, state, r` and has access to an encryption oracle
which, on input m
0
, m
1
, outputs c
pk
(m
b
∗ ; ω
∗
) for randomb
∗
and ω
∗
. Note that the value state deﬁnes
a key VK which is used by o1´
2
when giving its simulated proof. The values pk, σ, r
′
, state, and r
are ﬁxed throughout the simulation. When we say the simulator interacts with / using 'q, ω, b` we
mean that the simulator runs / as in Expt
1
; that is, encryption oracle query m
0
, m
1
is answered by
encrypting m
b
using randomness ω and then running o1´
2
(state; r), and the challenge sent by the
i
th
instance of the decryption oracle is q
i
. We note that decryption requests cannot be immediately
satisﬁed; this will not be a problem, as we show below.
We begin with simulation of the ﬁrst instance. The simulator chooses randomq, ω, b and interacts
with / using 'q, ω, b`. If / makes a call m
0
, m
1
to the encryption oracle before q
1
is requested, the
simulator forwards m
0
, m
1
to its encryption oracle and receives in return a ciphertext
¯
C
∗
; we then
say the ciphertext is deﬁned at instance 1. Once the ciphertext is deﬁned, the simulator no longer
needs to choose values ω, b and, in eﬀect, interacts with / using 'q, ω
∗
, b
∗
`. If the veriﬁcation key
VK
1
used by / in the ﬁrst instance is equal to VK deﬁned by state, the ﬁrst instance is declared
conditionally delinquent and the simulator proceeds to simulation of instance 2.
If the ciphertext is not deﬁned at instance 1, the simulator interacts with / using 'q, ω, b` until
the ﬁrst instance either succeeds or conclusively fails. Note that decryption of ciphertexts
¯
C
i
with
i > 1 is not required since such a request would imply that time β has elapsed since the sending of the
second message of instance i, but this would mean that time α has already elapsed since the second
message of instance 1 was sent (and therefore the ﬁrst instance has either succeeded or failed by that
point). If the ﬁrst instance succeeds, the simulator proceeds with witness extraction as described
below. If the ﬁrst instance fails, the simulator chooses new, random q, ω, b and interacts with / using
'q, ω, b`. This is repeated for a total of at most t
4
/τ times (using new, random q, ω, b each time) or
117
until the ﬁrst instance succeeds, where τ = τ(k) is an inverse polynomial whose value we will ﬁx at
the end of the proof and t = t(k) is a bound on the number of times / interacts with the decryption
oracle. If the ﬁrst instance ever succeeds, the simulator proceeds with witness extraction. Otherwise,
the ﬁrst instance is declared conditionally delinquent and the simulator proceeds to simulation of
the second instance.
If the ciphertext is deﬁned at instance 1, the simulator proceeds as above, but uses values ω
∗
, b
∗
to interact with / (where these values are deﬁned by ciphertext
¯
C
∗
received from the simulator’s
encryption oracle, as discussed previously).
If the ﬁrst instance ever succeeds, witness extraction will be performed. Assume the ﬁrst instance
succeeded when interacting with / using 'q, ω, b`. The simulator then does the following:
For n = 0 to e −1:
q
′
1
←Z
e
Interact with / using 'q
′
1
, q
2
, . . . , q
t
, ω, b`
If the ﬁrst instance succeeds and q
′
1
= q
1
, output the transcript and stop
Interact with / using 'n, q
2
, . . . , q
t
, ω, b`
if the ﬁrst instance succeeds and n = q
1
, output the transcript and stop
Output ⊥ and stop
If ⊥ is not output, the simulator attempts to compute a witness to the decryption of
¯
C
1
as in the
proof of Theorem 5.2. If such a witness is computed, we say the ﬁrst instance is extracted and the
simulator proceeds with simulation of instance 2. If ⊥ is output, or if ⊥ is not output but a witness
to the decryption of
¯
C
1
cannot be computed, the entire simulation is aborted; we call this a failure
to extract.
In general, when we are ready to simulate the i
th
instance (assuming the entire simulation has not
been aborted), each of the ﬁrst i −1 instances has been classiﬁed as either extracted or conditionally
delinquent. If instance j is extracted, the simulator knows the decryption of
¯
C
j
and can send it to /
upon successful completion of that instance in the current simulation. On the other hand, if instance
j is classiﬁed as conditionally delinquent, then with suﬃciently high probability that instance will
never succeed.
We say the ciphertext is deﬁned before instance i if, for some j ≤ i, the ciphertext is deﬁned at j.
At the beginning of simulation of the i
th
instance, if the ciphertext is not deﬁned before instance i−1,
the simulator chooses random q
i
, . . . , q
t
, ω, b and interacts with / using 'q
∗
1
, . . . q
∗
i−1
, q
i
, . . . , q
t
, ω, b`.
If / makes a call m
0
, m
1
to the decryption oracle before q
i
is requested, the simulator forwards
m
0
, m
1
to its encryption oracle and receives in return a ciphertext
¯
C
∗
; we then say the ciphertext is
deﬁned at instance i. If the veriﬁcation key VK
i
used by / during the i
th
instance of the decryption
oracle is equal to VK deﬁned by state, the i
th
instance is declared conditionally delinquent and the
simulator proceeds with simulation of the next instance.
118
If the ciphertext is not deﬁned before instance i, the simulator continues to interact with /
using 'q
∗
1
, . . . q
∗
i−1
, q
i
, . . . , q
t
, ω, b` until the i
th
instance either succeeds or conclusively fails. If a
success occurs, witness extraction is performed as described below. In case the i
th
instance fails, the
simulator chooses new, randomq
i
, . . . , q
t
, ω, b and interacts with /using 'q
∗
1
, . . . , q
∗
i−1
, q
i
, . . . , q
t
, ω, b`.
This is repeated for a total of at most t
4
/τ times or until the i
th
instance succeeds. If the i
th
instance
ever succeeds, the simulator proceeds with witness extraction as described below. Otherwise, the i
th
instance is declared conditionally delinquent and the simulator proceeds to simulation of the next
instance.
Note that during simulation of instance i, decryption of ciphertexts C
j
with j > i is not required
since such a request would imply that time β has elapsed since the sending of the second message
of instance j, but this would mean that time α has already elapsed since the second message of
instance i was sent (and therefore the i
th
instance has either succeeded or failed by that point).
However, the simulator may be required to decrypt ciphertext
¯
C
j
with j < i. In case instance j is
extracted, this is no problem, since the simulator knows the witness to the decryption of
¯
C
j
. On
the other hand, when j is conditionally delinquent, there is a problem. We handle this as follows:
if conditionally delinquent instance j succeeds before q
i
is sent, the entire simulation is aborted; we
call this a classiﬁcation failure. If a conditionally delinquent instance j succeeds after q
i
is sent, we
consider this an exceptional event at instance j during simulation of i, and do not include it in the
count of failed trials. However, if 3t
3
/τ such exceptional events occur for any j, the entire simulation
is aborted; we call this an exception at j during simulation of i.
If the ciphertext is deﬁned before instance i, the simulator proceeds as above but using ω
∗
, b
∗
(where these values are deﬁned by the ciphertext C
∗
obtained from the simulator’s encryption oracle,
as discussed previously).
If the i
th
instance ever succeeds, witness extraction will be performed. Assume the ﬁrst instance
succeeded when interacting with / using 'q, ω, b`. The simulator then does the following (success
here means that the i
th
instance succeeds and no exceptional events occurred during the interaction
with /):
For n = 0 to e −1:
q
′
i
←Z
e
Interact with / using 'q
1
, q
i−1
, q
′
i
, q
i+1
, . . . , q
t
, ω, b`
If the ﬁrst instance succeeds and q
′
i
= q
i
, output the transcript and stop
Interact with / using 'q
1
, q
i−1
, n, q
i+1
, . . . , q
t
, ω, b`
if the ﬁrst instance succeeds and n = q
i
, output the transcript and stop
Output ⊥ and stop
If ⊥ is not output, the simulator attempts to compute a witness to the decryption of
¯
C
i
as in
the proof of Theorem 5.2. If such a witness is computed, we say the i
th
instance is extracted and
119
proceed with simulation of the next instance. If ⊥ is output, or if ⊥ is not output but a witness to
the decryption of
¯
C
i
cannot be computed, the entire simulation is aborted; we call this a failure to
extract.
Once all t instances have been simulated, if the ciphertext is not deﬁned before t, the simulator
simply interacts with / on input 'q
∗
, ⊥, ⊥` until / makes call m
0
, m
1
to the encryption oracle. The
simulator forwards these values to its encryption oracle, receiving in return
¯
C
∗
. Simulation of the
encryption oracle for / is done using o1´
2
, as above. In this case, we say the ciphertext is deﬁned
at t + 1.
As long as the entire simulation is not aborted, the result is a perfect simulation of the view of /
in Expt
1
with random variables Ω
∗
def
= 'pk, σ, r
′
, state, r, w
∗
, b
∗
`. We now show that: (1) the expected
running time of the above simulation is polynomial in t and 1/τ, and (2) with all but negligible
probability over Ω
∗
and for some negligible function µ(), the simulation fails with probability at
most 3τ/4 + µ(k). Fixing τ(k) to an appropriate inverse polynomial function then yields a correct
simulation with suﬃciently high probability.
Claim 5.7 The expected running time of the simulation is polynomial in t and 1/τ.
For simulation of each instance, at most 4t
4
/τ trials are run before either aborting, declaring the
instance conditionally delinquent, or attempting to extract. Say extraction is attempted at instance
i because the i
th
instance succeeded (and no exceptional events occurred) when interacting with
/ using 'q, ω, b`. Let p
i
denote the probability, over choice of q
′
i
, that the i
th
instance succeeds
and no exceptional events occur when interacting with / using 'q
1
, . . . , q
i−1
, q
′
i
, q
i+1
, . . . , q
t
, ω, b`. If
p
i
> 1/e, extraction requires expected number of steps at most 2/p
i
. Since extraction with these
values of q, ω, b is performed with probability at most p
i
(4t
4
/τ), the contribution to the expected
running time is at most (4t
4
p
i
/τ) 2/p
i
= 8t
4
/τ. If p
i
≤ 1/e, extraction requires at most e steps,
and the contribution to the expected running time is then at most (4t
4
/eτ) e = 4t
4
/τ. In either
case, the contribution to the expected running time for simulation of any instance is polynomial
in t and 1/τ. Since there are at most t instances, the entire simulation has expected running time
polynomial in t and 1/τ.
Claim 5.8 With all but negligible probability over Ω
∗
, the probability of a classiﬁcation failure is at
most τ/4 +ε
4
(k), where ε
4
() is negligible.
The analysis follows [49]. Say the ciphertext is deﬁned at instance v, where 1 ≤ v ≤ t +1. For i < v,
deﬁne the values deﬁned before i as q
∗
1
, . . . , q
∗
i−1
and the variables not deﬁned before i as variables
q
i
, . . . , q
t
, ω, b; for i ≥ v, deﬁne the values deﬁned before i as q
∗
1
, . . . , q
∗
i−1
, ω
∗
, b
∗
and the variables
120
not deﬁned before i as q
i
, . . . , q
t
. For each i, recursively deﬁne d
i
as the probability (over variables
not deﬁned before i) that instance i succeeds, conditioned on the values deﬁned before i and on the
event that no delinquent instance j (j < i) succeeds. Deﬁne an instance i to be delinquent if d
i
is at
most τ/4t
3
. We now compute the probability that an instance which is not delinquent is classiﬁed
as conditionally delinquent.
If an instance i is declared conditionally delinquent because VK
i
= VK, then, with all but
negligible probability over Ω
∗
, the probability that instance i succeeds is negligible. If not, the
security of the one-time signature scheme is violated with non-negligible probability during Expt
1
(details omitted). If an instance is declared conditionally delinquent for failing too many trials, then,
if d
i
> τ/4t
3
, the probability that none of the t
4
/τ trials succeeded is at most

1 −
τ
4t
3

t
4
/τ
≤ e
−t/4
,
which is negligible.
Assuming that all instances declared conditionally delinquent are in fact delinquent, the proba-
bility of a classiﬁcation failure during a given instance is at most τ/4t
3
, and hence the probability
of a classiﬁcation failure occurring is at most t τ/4t
3
≤ τ/4.
Claim 5.9 With all but negligible probability over Ω
∗
, the probability of a failure to extract is negli-
gible.
A failure to extract occurs for one of two reasons: (1) the extraction algorithm cannot generate
two diﬀerent accepting transcripts or (2) the extraction algorithm generates two diﬀerent accepting
transcripts but cannot extract a witness to the decryption of the relevant ciphertext. Say extraction
is attempted at instance i because the i
th
instance succeeded (and no exceptional events occurred)
when interacting with / using 'q, ω, b`. Let p
i
denote (as in Claim 5.7) the probability, over choice
of q
′
i
, that the i
th
instance succeeds and no exceptional events occur when interacting with / using
'q
1
, . . . , q
i−1
, q
′
i
, q
i+1
, . . . , q
t
, ω, b`. If case (1) occurs, this implies that p
i
= 1/e. But in this case,
extraction at this instance and with these values of q, ω, b is performed only with probability at
most (4t
4
/τ) (1/e), which is negligible. Furthermore, since VK
i
= VK (otherwise instance i is
declared conditionally delinquent), the techniques of the proof of Theorem 5.3 imply that, with all
but negligible probability over Ω
∗
, case (2) occurs with only negligible probability. (If not, an RSA
root may be extracted in expected polynomial time with non-negligible probability.)
Claim 5.10 With all but negligible probability over Ω
∗
, the probability of abort due to an exception
at j during simulation of i (for any i, j) is at most τ/2 +ε
5
(k), where ε
5
() is negligible.
121
Fix i and j with i > j. Deﬁne d
′
i,j
as the probability (over variables not deﬁned before i) that instance
j succeeds (conditioned on the values deﬁned before i). An exception at j during simulation of i
means that, during simulation of i, conditionally delinquent instance j succeeded at least 3t
3
/τ times
out of at most 4t
4
/τ trials. We claim that if this happens, then, with all but negligible probability,
d
′
i,j
is at least 1/2t. If this were not the case, letting X be a random variable denoting the number
of successes in 4t
4
/τ trials and µ denote the actual value of d
′
i,j
, the Chernoﬀ bound shows that:
Pr[X >
3
2
µ4t
4
/τ] < Pr[X > 3t
3
/τ]
< (e/(3/2)
3
)
t
3
/τ
,
and since e/(3/2)
3
< 1, this expression is negligible. Assuming instance j is in fact delinquent (which
is true with all but negligible probability; see the proof of Claim 5.8), the probability (over variables
not deﬁned before j) that d
′
i,j
≥ 1/2t is at most τ/2t
2
. Summing over all t
2
possible choices of i and
j yields the desired result.
Assume the advantage of / in Expt
1
is not negligible. This implies the existence of some constant
c such that, for inﬁnitely many values of k,

−
3
2
(
1
2k
c
+µ(k)),
and then for inﬁnitely many values of k we have Adv
A
′ (k) ≥ 1/8k
c
so that this quantity is not
negligible. This completes the proof of the Theorem.
For completeness, we state the following theorems (in each case, we must also assume the security
of (SigGen, Sign, Vrfy) as a one-time signature scheme and the security of UOWH as a universal one-
way hash family):
Theorem 5.7 Assuming the hardness of factoring Blum integers for expected-polynomial-time algo-
rithms, the protocol of Figure 5.5 (with t = Θ(k)) is an interactive encryption scheme secure against
sequential chosen-ciphertext attacks. If timing constraints are enforced, the resulting protocol is
secure against concurrent chosen-ciphertext attacks.
122
Theorem 5.8 Assuming the hardness of the decisional
7
composite residuosity problem for expected-
polynomial-time algorithms, the protocol of Figure 5.6 is an interactive encryption scheme secure
against sequential chosen-ciphertext attacks. If timing constraints are enforced, the resulting protocol
is secure against concurrent chosen-ciphertext attacks.
Theorem 5.9 Assuming the hardness of the DDH
8
problem for expected-polynomial-time algo-
rithms, the protocol of Figure 5.7 is an interactive encryption scheme secure against sequential
chosen-ciphertext attacks. If timing constraints are enforced, the resulting protocol is secure against
concurrent chosen-ciphertext attacks.
These schemes are in fact quite eﬃcient, especially in comparison with the underlying semantically
secure schemes. For example, the RSA-based scheme of Figure 5.3 for encryption of ℓ-bit messages
requires only 2ℓ+4 exponentiations compared to the ℓ exponentiations required by the basic scheme.
Furthermore, the additional exponentiations may be done in a preprocessing stage before the message
to be sent is known.
5.4.2 Password-Based Authentication and Key Exchange
Password-based authentication and key exchange in the public-key setting were ﬁrst formally mod-
eled by Halevi and Krawczyk [76]; Boyarsky [22] extends the model to the multi-party setting (the
model and deﬁnition of security for this setting are essentially identical to that presented in Chapter
3, except that the adversary is now additionally given the public keys of the servers). Protocols for
password-based authentication may be constructed from any chosen-ciphertext-secure encryption
scheme [76, 22]. Let pw be the password of the user which is stored by the server. A password-based
authentication protocol using a (non-interactive) chosen-ciphertext-secure encryption scheme has the
server send a random, suﬃciently-long nonce n to the user, who replies with an encryption of pw◦ n
(actually, this brief description suppresses details which are unimportant for the discussion which
follows; see [76, 22]). The server decrypts and veriﬁes correctness of the password and the nonce;
the nonce is necessary to prevent replay attacks. When an interactive, chosen-ciphertext-secure
encryption scheme is used, the nonce is not necessary if the probability that a server repeats its
messages is negligible [22]. In this case, authentication proceeds by simply having the user perform
a (random) encryption of pw.
Password-based key-exchange protocols (with or without mutual authentication) may also be
constructed using any chosen-ciphertext-secure encryption scheme [76, 22]. Here, for example, the
7
As mentioned in Section 5.3.3, a construction secure under the (weaker) computational variant of this assumption
is also possible.
8
As mentioned in Section 5.3.4, a construction secure under the (weaker) CDH assumption is also possible.
123
user responds to a random nonce n with an encryption of pw◦ n◦ k, where k is the key to be shared
(this achieves one-way authentication only; mutual authentication can be achieved with an additional
round). As above, when an interactive chosen-ciphertext-secure encryption scheme is used, the nonce
is not necessary when the probability of repeat messages from the server is negligible.
The only previously-known eﬃcient and provably-secure implementations use the public-key en-
cryption scheme of [36] whose security relies on the DDH assumption (the interactive solution of [44]
may also be used, but, as mentioned previously, the solutions presented here are more eﬃcient). Our
techniques allow eﬃcient implementation of these protocols based on a wider class of assumptions.
For the chosen-ciphertext-secure schemes given in the previous section, the probability that
a server (acting as a receiver for the given encryption scheme) repeats a challenge is negligible.
Figures 5.3, 5.5–5.7 therefore immediately yield eﬃcient, 3-round, password-based authentication or
key-exchange protocols in the public-key model. Security of these protocols may be based on the
hardness of factoring, the RSA problem, or the decisional composite residuosity assumption. We
stress that these are the ﬁrst eﬃcient and provably-secure constructions based on assumptions other
than DDH.
5.4.3 Deniable Authentication
Previous work. Deniable authentication was ﬁrst considered in [44], and a formal deﬁnition appears
in [49]. The strongest notion of security requires the existence of a simulator which, given access
only to a malicious veriﬁer, can output a transcript which is indistinguishable from an interaction
of the veriﬁer with the actual prover. Constructions based on any non-malleable encryption scheme
are known [49, 50, 48]. However, these protocols are not secure (in general) when a non-malleable
interactive encryption scheme is used. For example, the non-malleable, interactive encryption scheme
of [44] requires a signature from the prover and hence the resulting deniable authentication protocol
is not simulatable (this problem is pointed out explicitly by Dwork, et al. [49]). Thus, the only
previously-known, eﬃcient deniable-authentication protocol which is secure under the strongest
deﬁnition of security uses the construction of Dwork et al. [49] instantiated with the Cramer-Shoup
encryption scheme [36]. Our constructions have the same round-complexity and eﬃciency, but their
security may be based on a larger (and, in some cases, weaker) class of assumptions.
Deﬁnitions. We begin with a review of the deﬁnition that appears in [49, 48]. We have a prover
{ who has established a public key using key-generation algorithm / and is willing to authenticate
messages to a veriﬁer 1; however, { is not willing to allow the veriﬁer to convince a third party
(after the fact) that { authenticated anything. This is captured by ensuring that a transcript of an
execution of the authentication protocol can be eﬃciently simulated without any access to {. We also
124
require that no malicious adversary will be able to impersonate {. More speciﬁcally, an adversary
´ (acting as man-in-the-middle between { and a veriﬁer) should not be able to authenticate a
message m to the veriﬁer which { does not authenticate for ´.
Deﬁnition 5.4 Let Π = (/, {, 1) be a tuple of ppt algorithms. We say Π is a strong deniable-
authentication protocol over message space M= ¦M
pk
¦
pk∈K(1
k
),k∈N
if it satisﬁes the following:
(Completeness) For all (pk, sk) output by /(1
k
) and all m ∈ M
pk
, we have '{
sk
(m), 1`(pk) = m
(when 1 does not output ⊥ we say it accepts).
(Soundness) Let ´ be a ppt adversary with oracle access to {, where { authenticates any poly-
nomial number of messages chosen adaptively by ´. Then the following is negligible in k:
Pr[(pk, sk) ← /(1
k
); m ← '´
P
sk
(·)
, 1` : m / ∈ ¦m
1
, . . . , m
ℓ
¦ ∧ m =⊥],
where m
1
, . . . , m
ℓ
are the messages authenticated by {.
(Strong deniability) Let 1
′
be a ppt adversary interacting with {, where { authenticates any
polynomial number of messages chosen adaptively by 1
′
. There exists an expected-polynomial-time
simulator o1´ with black-box (rewind) access to 1
′
such that, with all but negligible probability over
pk output by /(1
k
), the following distributions are statistically indistinguishable:
¦pk, o1´(pk)¦
¦pk, 1
′
P
sk
(pk)¦.
We also consider deniable-authentication protocols with a slightly weaker guarantee on their denia-
bility.
Deﬁnition 5.5 Let Π = (/, {, 1) be a tuple of ppt algorithms. We say Π is a strong ε-deniable-
authentication protocol over message space M= ¦M
pk
¦ if it satisﬁes completeness and soundness as
in Deﬁnition 5.4 in addition to the following:
(Strong ε-deniability) Let 1
′
be a ppt adversary interacting with {, where { authenticates any
polynomial number of messages chosen adaptively by 1
′
. There exists a negligible function µ() and
a simulator o1´ with black-box (rewind) access to 1
′
such that, for all ε > 0, the expected running
time of o1´(pk, ε) is polynomial in k and 1/ε and, with all but negligible probability over pk output
by /(1
k
), the following distributions have statistical diﬀerence at most ε +µ(k):
¦pk, o1´(pk, ε)¦

pk, 1
′
P
sk
(pk)
¸
.
125
A relaxation of the above deﬁnitions which has been considered previously (e.g., [49]) allows
the simulator o1´ to have access to {
sk
when producing the simulated transcript, but { only
authenticates some ﬁxed sequence of messages independent of those chosen by 1
′
. We call this weak
deniability. In practice, weak deniability may not be acceptable because the protocol then leaves an
undeniable trace that P authenticated something (even if not revealing what ). However, { may want
to deny that any such interaction took place. Note that previous solutions based on non-malleable
encryption [44, 49, 50, 48] only achieve weak deniability when using the interactive non-malleable
encryption scheme suggested by [44] (which requires a signature from {).
Constructions. The protocols of Section 5.3 may be easily adapted to give deniable-authentication
protocols whose security rests on the one-wayness of the appropriate encryption scheme for random
messages; semantic security of the encryption scheme is not necessary. This yields very eﬃcient
deniable-authentication protocols since, for example, we may use the “simple” RSA encryption
scheme in which r is encrypted as r
e
mod N (under the RSA assumption, this scheme is one-way
for random messages). Furthermore, eﬃcient deniable-authentication protocols may be constructed
using weaker assumptions; for example, using the CDH assumption instead of the DDH assumption.
Below, we improve the eﬃciency of these schemes even further, although in this case the resulting
protocols are only secure over polynomially-large message spaces.
We ﬁrst present the paradigm for construction of protocols which are secure over exponentially-
large message spaces. The basic idea is for the receiver to give a non-malleable PPK for a ciphertext
C encrypted using an encryption scheme which is one-way for random messages. Additionally,
the message m which is being authenticated is included in the transcript and is signed along with
everything else. Assuming the receiver’s proof succeeds, the prover authenticates the message by
responding with the decryption of C.
Figure 5.8 shows an example of this approach applied to the non-malleable PPK for RSA en-
cryption. The public key of the prover { is an RSA modulus N, a prime e (with [e[ = Θ(k)),
elements g, h ∈ Z
∗
N
, and a hash function H chosen randomly from a family of universal one-way
hash functions. Additionally, the prover has secret key d such that de = 1 mod ϕ(N). The veriﬁer
1 has message m taken from an arbitrary message space (of course, [m[ must be polynomial in the
security parameter). To have m authenticated by {, the veriﬁer chooses a random y ∈ Z
∗
N
, computes
C = y
e
, and then performs a non-malleable proof of knowledge of the witness y to the decryption of
C (as in Figure 5.3). Additionally, the message m is sent as the ﬁrst message of the protocol, and is
signed along with the rest of the transcript. If the veriﬁer’s proof succeeds, the prover computes C
d
and sends this value to the veriﬁer. If the proof does not succeed, the prover simply replies with ⊥.
As in the case of interactive encryption in Section 5.4.1, timing constraints are needed when
126
Public key: N; prime e; g, h ∈ Z
∗
N
; H : {0, 1}
∗
→ Ze
1 (input m ∈ ¦0, 1¦
∗
) { (input d)
(VK, SK) ← SigGen(1
k
)
y, r
1
, R
2
←Z
∗
N
; q
1
←Z
e
C := y
e
; α := H(VK)
A
1
:= r
e
1
A
2
:= R
e
2
/ (g
α
h)
q1
VK, m, C, A
1
, A
2
-
q
2
←Z
e
q
2

Figure 5.8: A deniable-authentication protocol based on RSA.
concurrent access to the prover is allowed. In this case, we require that the veriﬁer respond to the
challenge (i.e., send the third message of the protocol) within time α from when the challenge was
sent. If 1 does not respond within this time, the proof is rejected. Additionally, the last message of
the protocol is not sent by the prover until at least time β has elapsed since sending the challenge
(clearly, we must have β > α).
Theorem 5.10 Assuming (1) the hardness of the RSA problem for expected-polynomial-time algo-
rithms, (2) the security of (SigGen, Sign, Vrfy) as a one-time signature scheme, and (3) the secu-
rity of UOWH as a universal one-way hash family, the protocol of Figure 5.8 is a strong deniable-
authentication protocol, over an arbitrary message space, for adversaries given sequential access to
the prover. If timing constraints are enforced as outlined above, the protocol is a strong ε-deniable-
authentication protocol for adversaries given concurrent access to the prover.
Proof We assume familiarity with the proofs of Theorems 5.3 and 5.6. We consider the more
challenging case of concurrent access to the prover; results for the case of sequential access may be
derived from the arguments given here. Correctness of the protocol is immediate. We demonstrate
soundness of the protocol using the same techniques as in the proof of Theorem 5.6. Given a ppt
adversary ´, we construct an expected-polynomial-time adversary /
′
who attacks the one-wayness
of the encryption scheme for a random message. In particular, adversary /
′
takes as input modulus
127
N and a random C ∈ Z
∗
N
and attempts to compute y = C
1/e
mod N. Deﬁne Succ as the event
that ´ authenticates a message for 1 which was not authenticated by { (cf. Deﬁnition 5.4). We
show that the success probability of /
′
in inverting C will be negligible if and only if Pr[Succ] is
negligible. This immediately implies soundness of the deniable-authentication protocol.
Let t(k), which is polynomial in k, be a bound on the number of times ´accesses the prover when
run on security parameter 1
k
; without loss of generality, we assume that t(k) ≥ k (for convenience,
in the remainder of the proof we suppress the dependence on k and simply write t). Let pk denote
the components N, e of the public key and let σ denote components g, h, H. In the real experiment
Expt
0
, the actions of ´ are completely determined by pk
′
= 'pk, σ`, random coins r
′
for ´, the
vector of challenges q = q
1
, . . . , q
t
used during the t interactions of ´ with the prover, and the
randomness used by the veriﬁer (this includes the randomness y used to generate C as well as the
randomness used for execution of the PPK). Let Pr[Succ
0
] denote the probability of event Succ in
the real experiment.
We modify the real experiment giving Expt
1
, as follows. Component pk of the public key is
generated normally, along with the corresponding secret key sk; however, σ is generated using
o1´
1
(pk), which also generates state. The adversary ´ is run on input pk
′
= 'pk, σ`. The
adversary’s calls to the prover are handled as in Expt
0
(in particular, any ciphertext C may be
decrypted since sk is known), but the adversary’s call to the veriﬁer will be handled diﬀerently.
When ´ calls the veriﬁer on message m, we now compute C = y
e
for random y and simulate the
PPK for C (including m in the transcript) using algorithm o1´
2
(state; r) for randomly-chosen r.
Now, the actions of ´ are completely determined by pk
′
, random coins r
′
for ´, the vector of
challenges q = q
1
, . . . , q
t
used by the prover, the value y, and the values state and r used by o1´
2
in simulating the veriﬁer. Since o1´ yields a perfect simulation of a real execution of the PPK (cf.
Theorem 5.2), we have Pr[Succ
1
] = Pr[Succ
0
], where the ﬁrst probability refers to the probability of
event Succ in Expt
1
.
For the ﬁnal experiment Expt
2
, interaction with the prover will be handled by extracting witnesses
to the decryption of the various ciphertexts from the proofs given by ´(in particular, sk will not be
used). Let Ω
def
= 'pk, σ, r
′
, q, state, r` and let Ω
∗
def
= 'Ω, y`. We show below an expected-polynomial-
time simulator which takes Ω and C as input (where C = y
e
). For any particular Ω
∗
, let p
Ω
∗ be
the probability that the simulator succeeds in simulating the execution of ´ in Expt
1
with random
variables Ω
∗
. Then for some negligible function µ() and with all but negligible probability over Ω
∗
,
we will have p
Ω
∗ > 1/2 − µ(k). As in the proof of Theorem 5.6, this implies that Pr[Succ
2
] is at
least 1/2 Pr[Succ
1
] −ε
1
(k), for some negligible function ε
1
().
Before giving the details of the simulation, we note that Expt
2
immediately suggests an adversary
128
/
′
attacking the one-wayness of RSA encryption. Given a public key pk and a ciphertext C,
adversary /
′
runs o1´
1
(pk) to generate parameters σ and state. /
′
then ﬁxes the randomness r
′
of ´, and runs ´ on input pk
′
= 'pk, σ`. Simulation of the veriﬁer for ´ is done as follows: when
´ gives message m, adversary /
′
uses o1´
2
(state; r), for randomly-chosen r, to simulate a PPK
for C and includes m in the transcript. The decryption oracle will be simulated as in Expt
2
(details
of which appear below). /
′
outputs whatever value ´ sends to the veriﬁer as the ﬁnal message of
that interaction. Note that the probability that /
′
successfully outputs C
1/e
is exactly Pr[Succ
2
].
One-wayness of RSA implies that this is negligible.
Simulation of { in Expt
2
may be done exactly as in the proof of Theorem 5.6. In fact, the proof
here is even easier since the ciphertext C to be inverted is known to /
′
at the beginning of the
simulation; therefore, we do not need to worry about the instance at which C is deﬁned. In brief,
the simulator is given values 'pk, σ, r
′
, q
∗
, state, r, C` (where C = y
e
for some y unknown to the
simulator). The values pk, σ, r
′
, state, r, and C are ﬁxed throughout the simulation. When we say
the simulator interacts with ´ using 'q` we mean that the simulator runs ´ as in Expt
1
; that is, if
´ submits message m to 1, this query is answered by using o1´
2
(state; r) to simulate a PPK of
C and including m in the transcript, and the challenge sent by the i
th
instance of the prover is q
i
.
When we are ready to simulate the i
th
instance (assuming the entire simulation has not been
aborted), each of the ﬁrst i−1 instances has been classiﬁed as either extracted or conditionally delin-
quent. The simulator chooses random q
i
, . . . , q
t
and interacts with ´ using 'q
∗
1
, . . . , q
∗
i−1
, q
i
, . . . , q
t
`
until either (1) the i
th
instance succeeds or (2) the i
th
instance fails. If success occurs, witness
extraction is performed as in the proof of Theorem 5.6. Otherwise, the above process is repeated at
most t
4
/τ times; if the i
th
instance never succeeds, it is classiﬁed as conditionally delinquent.
The remaining details of the simulation may be derived from the proof of Theorem 5.6. This
completes the proof of soundness.
For the proof of strong ε-deniability, note that the proof of Theorem 5.6 actually gives, for all
τ > 0, a simulator which has expected running time polynomial in k and 1/τ; furthermore, with
all but negligible probability over Ω
∗
the simulation is aborted with probability at most τ plus a
negligible quantity. The modiﬁed simulator for the present context, as described above, inherits this
property. This immediately implies strong ε-deniability.
We brieﬂy sketch the proof of strong deniability for the sequential case (where timing constraints
are not used). Here, the simulator is given values pk
′
, r
′
, and q
∗
and must simulate the interaction of
a malicious veriﬁer 1
∗
with {. To do so, the simulator ﬁxes pk, σ, sets the random tape of 1
∗
to r
′
and gives pk
′
to 1
∗
as input, and interacts with 1
∗
using q
∗
. To simulate the i
th
instance (assuming
all previous instances have been simulated and the entire transcript has not been aborted), the
129
simulator interacts with 1
∗
using q
∗
until 1
∗
sends the third message of the i
th
instance. If this
instance fails, the simulator responds with ⊥ and continues with simulation of the next instance. If
this instance succeeds, the simulator runs the witness extraction procedure as in the proof of Theorem
5.6. If witness extraction fails, the simulation is aborted; we call this a failure to extract. If witness
extraction succeeds, the simulator now knows C
1/e
i
and can therefore continue with simulation of
the next instance.
If an instance j fails when interacting with 1
∗
using q
1
, . . . , q
j
, q
j+1
, . . . , q
t
then this instance
always fails when interacting with 1
∗
using q
1
, . . . , q
j
, q
′
j+1
, . . . , q
′
t
for arbitrary q
′
j+1
, . . . , q
′
t
; this
follows from the sequential access of 1
∗
. Therefore, no classiﬁcation failures or exceptional events
can occur. Thus, the simulation is aborted only following a failure to extract; however, as shown in
the proof of Theorem 5.6, the probability of a failure to extract is negligible.
For completeness, we state the following theorems (in each case, we must also assume the security
of (SigGen, Sign, Vrfy) as a one-time signature scheme and the security of UOWH as a universal one-
way hash family):
Theorem 5.11 Assuming the hardness of factoring Blum integers for expected-polynomial-time al-
gorithms, the protocol of Figure 5.5 may be adapted, as above, to give a strong deniable-authentication
protocol (over an arbitrary message space) for adversaries given sequential access to the prover. If
timing constraints are enforced, the protocol is an strong ε-deniable-authentication protocol for ad-
versaries given concurrent access to the prover.
Theorem 5.12 Assuming the hardness of the computational composite residuosity problem for
expected-polynomial-time algorithms, the protocol of Figure 5.6 may be adapted, as above, to give
a strong deniable-authentication protocol (over an arbitrary message space) for adversaries given
sequential access to the prover. If timing constraints are enforced, the protocol is a strong ε-deniable-
authentication protocol for adversaries given concurrent access to the prover.
Theorem 5.13 Assuming the hardness of the computational Diﬃe-Hellman problem for expected-
polynomial-time algorithms, the protocol of Figure 5.7 may be adapted, as above, to give a strong
deniable-authentication protocol (over an arbitrary message space) for adversaries given sequen-
tial access to the prover. If timing constraints are enforced, the protocol is a strong ε-deniable-
authentication protocol for adversaries given concurrent access to the prover.
We stress that these protocols are quite practical. For example, the protocol implied by Theorem
5.13 has the same round-complexity, requires fewer exponentiations, has a shorter public key, and is
based on a weaker assumption than the most eﬃcient, previously-known protocol for strong deniable
130
Public key: G; g, h0, h1 ∈ G
1 (input m ∈ M
q
⊂ Z
q
) { (input x = log
g
h
0
)
y, r
0
, z
1
, c
2
←Z
q
C := g
y
A
1
:= g
r0
A
2
:= g
z1
/ (h
m
0
h
1
)
c2
m, C, A
1
, A
2
-
c ←Z
q
c

Figure 5.9: A deniable-authentication protocol for polynomial-size message spaces.
authentication (i.e., the protocol of [49] instantiated with the Cramer-Shoup encryption scheme [36]).
No previous eﬃcient protocols were known based on the RSA, factoring, or computational/decisional
composite residuosity assumptions.
The eﬃciency of the above constructions may be improved further; however, the resulting pro-
tocols are secure over polynomial-sized message spaces only.
9
In Figure 5.9, we illustrate the im-
provement for the non-malleable PPK of Figure 5.7 (similar modiﬁcations to Figures 5.3, 5.5, and
5.6 yield protocols whose security may be based on the RSA, factoring, or computational composite
residuosity assumptions). In the improved protocol, we make use of the fact that the adversary
cannot re-use the value m which the adversary attempts to authenticate for the veriﬁer (since, to
break the security of the scheme, the adversary must authenticate a message which the prover does
not authenticate). This eliminates the need to use a veriﬁcation key VK, and eliminates the need
for a one-time signature on the transcript. However, since the adversary chooses m (and this value
must be guessed by the simulator in advance; see proof below), the present schemes are only secure
when the message space is polynomial-size.
Theorem 5.14 Assuming the hardness of the CDH problem for expected-polynomial-time algo-
rithms, the protocol of Figure 5.9 is a strong deniable-authentication protocol (over message space
¦M
q
¦ where M
q
⊂ Z
q
and [M
q
[ is polynomial in k) for adversaries given sequential access to the
9
In Section 5.4.5, we modify the protocol based on the CDH assumption to achieve security over an exponentially-
large message space.
131
prover. If timing constraints are enforced, the protocol is a strong ε-deniable-authentication protocol
for adversaries given concurrent access to the prover.
Proof Completeness is obvious. Strong deniability in the sequential case and strong ε-deniability
in the concurrent case follow from the arguments in the proof of Theorem 5.10. We prove soundness
of the protocol for the more diﬃcult, concurrent case.
Given a ppt adversary ´, we construct an expected-polynomial-time algorithm A which solves
an instance of the CDH problem. Let Succ be the event that ´ authenticates a message for 1
which was not authenticated by { (cf. Deﬁnition 5.4). We show that the success probability of A in
solving a random CDH instance is negligible if and only if Pr[Succ] is negligible. This immediately
implies soundness of the deniable-authentication protocol.
Let the size of M
q
be p(k) (where p() is polynomial in the security parameter), and let t(k)
(where t() is polynomial in k), be a bound on the number of times ´ accesses the prover when run
on security parameter 1
k
; without loss of generality, we assume that t(k) ≥ k (for convenience, in
the remainder of the proof we suppress the dependence on k and simply write p, t). Let pk denote
the values G, g, h
0
, h
1
which constitute the public key. In the real experiment Expt
0
, the actions of
´ are completely determined by pk, random coins r
′
for ´, the vector of challenges c = c
1
, . . . , c
t
used during the t interactions of ´ with the prover, and the randomness used by the veriﬁer (this
includes the randomness y used to generate C as well as the randomness ω
r
used for the real PPK).
Let Pr[Succ
0
] denote the probability of event Succ in the real experiment.
We next modify the real experiment, giving Expt
1
, as follows. Public key pk is generated by
choosing g at random in G, choosing random m
∗
∈ M and x, r ∈ Z
q
, and setting h
0
= g
x
and
h
1
= h
m
∗
0
g
r
. The adversary ´ is run on input G, g, h
0
, h
1
. The adversary’s calls to the prover are
handled as in Expt
0
(this can be done eﬃciently since log
g
h
0
is known), but the adversary’s call
to the veriﬁer will be handled diﬀerently. If ´ ever asks the prover to authenticate message m
∗
,
abort. Furthermore, if ´ calls the veriﬁer on message m with m = m
∗
, abort. (Without loss of
generality, we assume that if ´ calls the veriﬁer on message m
∗
, then ´ never asks the prover
to authenticate m
∗
; note that ´, by deﬁnition, cannot succeed if this occurs.) If m = m
∗
, choose
C
∗
∈ G at random, and simulate a PPK for C
∗
as in the proof of Theorem 5.5. In particular,
choose z
0
, r
1
, c
1
∈ Z
q
at random, set A
1
= g
z0
/(C
∗
)
c1
and A
2
= g
r1
, and send m
∗
, C
∗
, A
1
, A
2
as
the ﬁrst message. Upon receiving query c from ´, set c
2
= c − c
1
and compute z
1
= c
2
r + r
1
; the
values c
1
, z
0
, z
1
are sent as the response. Note that the actions of ´ are now completely determined
by pk, r
′
, c, C
∗
, ω
f
, where ω
f
denotes the randomness used for the simulated PPK. Let Pr[Succ
1
]
denote the probability of event Succ in this experiment. Since pk hides all information about the
choice of m
∗
, the probability of not aborting is exactly 1/p; furthermore, since the simulated proof
132
is identically-distributed to a real proof, we have Pr[Succ
1
] = 1/p Pr[Succ
0
].
For the ﬁnal experiment Expt
2
, actions of the prover will be simulated by extracting log
g
C
i
from the proofs given by ´ in its various interactions with the prover (in particular, log
g
h
0
will
not be used). Let Ω
def
= 'pk, r
′
, c, C
∗
, ω
f
`. We show below an expected-polynomial-time simulator
which takes Ω as input; furthermore, for all Ω and some negligible ε
1
(), the simulator succeeds in
simulating the execution of ´in Expt
1
with random variables Ω with probability at least 1/2−ε
1
(k).
Therefore, Pr[Succ
2
] > 1/2 Pr[Succ
1
] −ε
1
(k). In particular (recalling that p is polynomial in k), if
Pr[Succ
0
] is non-negligible, then so is Pr[Succ
2
].
Before giving the details of the simulation, we note that Expt
2
immediately suggests an adversary
A which solves the CDH problem. Given g, h
0
, C
∗
as input, A ﬁxes random r
′
, c, and ω
f
, and runs
´ as in Expt
2
. Note that this can be eﬃciently done (given the simulator we describe below); in
particular log
g
h
0
is not needed. Finally, A outputs whatever value ´ sends to the veriﬁer as the
ﬁnal message of that interaction. The probability that A solves the given CDH instance is exactly
Pr[Succ
2
]. Hardness of the CDH problem implies that this is negligible.
It remains to show how to simulate the actions of the prover. Deﬁne the i
th
instance of the prover
as the i
th
time that / requests the second message of the PPK (i.e., the challenge) be sent by the
prover. In any transcript of the execution of ´, we let C
i
denote the second component of the ﬁrst
message of the i
th
instance of the prover; deﬁne m
i
similarly. For any instance of the prover, we say
the instance succeeds if (1) an honest prover would accept the instance, (2) the timing constraints
are satisﬁed for that instance. Otherwise, we say the instance fails.
The simulator is given values 'pk, r
′
, c
∗
, C
∗
, ω
∗
f
`; furthermore, the value m
∗
is as deﬁned above for
Expt
1
. The values pk, r
′
are ﬁxed throughout the simulation. When we say the simulator interacts
with ´ using 'c, y, ω
r
, C, ω
f
` we mean the following:
• If C, ω
f
=⊥ and y, ω
r
=⊥, the simulator runs ´using c as the challenges of the prover. When
´ interacts with the veriﬁer, the simulator runs the honest protocol for the veriﬁer, sending
C = g
y
and executing a real PPK using coins ω
r
.
• If y, ω
r
=⊥ and C, ω
f
=⊥, the simulator runs ´using c as the challenges of the prover. When
´ interacts with the veriﬁer, the simulator sends C
∗
and executes the simulated PPK using
coins ω
f
. Note that such a simulation is only possible if m
∗
is the message ´ sends to the
veriﬁer.
Decryption requests cannot be immediately satisﬁed; this will not be a problem, as we show below.
To begin, the simulator chooses random q, y, ω
r
and interacts with ´ using 'c, y, ω
r
, ⊥, ⊥`. If
´ sends message m to the veriﬁer and m = m
∗
, the simulation is aborted. If ´ sends message m
133
to the veriﬁer and m = m
∗
, we say the message is deﬁned at instance 1. Otherwise, the message is
not deﬁned at instance 1.
If the message is not deﬁned at instance 1, the simulator interacts with ´ using 'c, y, ω
r
, ⊥, ⊥`.
If m
1
= m
∗
, the simulation is aborted. Otherwise, interaction continues until either (1) the ﬁrst
instance succeeds or (2) the ﬁrst instance fails. Note that the simulator cannot yet simulate the
prover for instances j > i; however, this is not required, since a request for such a simulation implies
that the ﬁrst instance has already failed due to the timing constraints.
If the ﬁrst instance ever succeeds, witness extraction is performed as described below. In case
the ﬁrst instance fails, the simulator chooses new, random c, y, ω
r
and interacts with ´ using
'c, y, ω
r
, ⊥, ⊥`. This is repeated at most t
4
/τ times (using new, random q, ω, b each time) or until
the ﬁrst instance succeeds, where τ is a constant whose value we will ﬁx at the end of the proof. If
the ﬁrst instance ever succeeds, the simulator proceeds with witness extraction as described below.
Otherwise, the ﬁrst instance is declared conditionally delinquent and the simulator proceeds to
simulation of the second instance as described below.
If the message is deﬁned at instance 1, the simulator proceeds as above, but interacts with ´
using 'c, ⊥, ⊥, C
∗
, ω
∗
f
`.
If the ﬁrst instance ever succeeds, witness extraction will be performed. Assume the ﬁrst instance
succeeded when interacting with ´ using 'c, y, ω
r
, C, ω
f
`. The simulator does the following:
For n = 0 to q −1:
c
′
1
←Z
q
Interact with ´ using 'c
′
1
, c
2
, . . . , c
t
, y, ω
r
, C, ω
f
`
If the ﬁrst instance succeeds and c
′
1
= c
1
, output the transcript and stop
If g
n
= C
1
output n and stop
If a value in Z
q
is output, we immediately have y
1
(= log
g
C
1
). If a second transcript is output, the
simulator computes (cf. the proof of Theorem 5.5) either y
1
or log
g
(h
m1
0
h
1
) for m
1
= m
∗
(recall the
simulation is aborted if m
i
= m
∗
is ever sent by ´ during any of the instances). In this second
case, the simulator may then compute x = log
g
h
0
. In either case, the simulator can now simulate
the ﬁrst instance by responding with either C
x
1
or h
y1
0
and we say the instance is extracted.
In general, when we are ready to simulate the i
th
instance (assuming the entire simulation has not
been aborted), each of the ﬁrst i −1 instances has been classiﬁed as either extracted or conditionally
delinquent. If instance j is extracted, then the simulator can send the appropriate value to ´ upon
successful completion of that instance in the current simulation. On the other hand, if instance j is
classiﬁed as conditionally delinquent, then with suﬃciently high probability that instance will never
succeed.
We say the message is deﬁned before instance i if, for some j ≤ i, the message is deﬁned
at instance j. At the beginning of simulation of the i
th
instance, if the message is not deﬁned
134
before instance i − 1, the simulator chooses random c
i
, . . . , c
t
, y, ω
r
and interacts with ´ using
'c
∗
1
, . . . c
∗
i−1
, c
i
, . . . , c
t
, y, ω
r
, ⊥, ⊥`. If ´ sends message m to the veriﬁer and m = m
∗
, the simula-
tion is aborted. If ´ sends message m to the veriﬁer and m = m
∗
, we say the message is deﬁned at
instance i. Otherwise, the message is not deﬁned at instance i.
If the message is not deﬁned before instance i, the simulator continues to interact with ´ using
'c
∗
1
, . . . c
∗
i−1
, c
i
, . . . , c
t
, y, ω
r
, ⊥, ⊥`. If m
i
= m
∗
, the simulation is aborted. Otherwise, interaction
continues until either: (1) the i
th
instance succeeds or (2) the i
th
instance fails. This is repeated for
a total of at most t
4
/τ times or until the i
th
instance succeeds. If the i
th
instance ever succeeds,
the simulator proceeds with witness extraction similar to what was described above (details omit-
ted). Otherwise, the i
th
instance is declared conditionally delinquent and the simulator proceeds to
simulation of the next instance.
Note that during simulation of instance i, simulation of instance j with j > i is not required since
such a request would imply that time the i
th
instance has already failed due to timing constraints.
However, the simulator may be required to simulate instances j with j < i. In case instance j
is extracted, this is not problem. On the other hand, if j is conditionally delinquent, simulation
cannot continue. We handle this as follows: if conditionally delinquent instance j succeeds before
c
i
is sent, the entire simulation is aborted; we call this a classiﬁcation failure. If a conditionally
delinquent instance j succeeds after c
i
is sent, we consider this an exceptional event at instance j
during simulation of i, and do not include it in the count of failed trials. However, if 3t
3
/τ such
exceptional events occur for any j, the entire simulation is aborted; we call this an exception at j
during simulation of i.
If the message is deﬁned before instance i, the simulator proceeds as above, but interacts with
´ using 'c
∗
1
, . . . , c
∗
i−1
, c
i
, . . . , c
t
, ⊥, ⊥, C
∗
, ω
∗
f
`.
Once all t instances have been simulated in this way, if the message is not deﬁned before instance
t, then the simulator interacts with ´ using 'c
∗
, ⊥, ⊥, C
∗
, ω
∗
f
` until completion.
Claim 5.11 The expected running time of the simulation is polynomial in t and 1/τ.
The proof is as for Claim 5.7.
For the following two claims, it is important to note that for any c, the distribution of actions of
´when the simulator interacts with ´using 'c, y, ω
r
, ⊥, ⊥` (for random choice of y, ω
r
) is identical
to the distribution of actions of ´ when the simulator interacts with ´ using 'c, ⊥, ⊥, C, ω
f
` (for
random choice of C, ω
f
). This follows from the fact that the simulated PPK is distributed identically
to a real PPK.
Claim 5.12 The probability of a classiﬁcation failure is at most τ/4+ε
2
(k), where ε
2
() is negligible.
135
The proof is as for Claim 5.8, except that the claim holds for all Ω since the adversary is assumed
not to ask the prover to authenticate m
∗
once that value is given to the veriﬁer (in Claim 5.8, ´
was “prevented” from re-using VK by virtue of the security of the signature scheme).
Claim 5.13 The probability of abort due to an exception at j during simulation of i (for any i, j) is
at most τ/2 +ε
3
(k), where ε
3
() is negligible.
The proof is as in Claim 5.10, except that, as above, the claim holds for all Ω.
Note that the probability of a failure to extract is 0 in this case; however, for the protocols based
on RSA, factoring, or Paillier, this probability would be negligible with all but negligible probability
over Ω.
As in the proof of Theorem 5.6, setting τ = 2/3 yields the desired simulation.
5.4.4 Identiﬁcation
Deﬁnitions and preliminaries. Identiﬁcation protocols satisfying various notions of security are
known [56, 111, 74, 100, 31]. Only recently, however, has a deﬁnition of security against man-in-the-
middle attacks been given by Bellare, et al. [7]. They provide practical, 4-round protocols and less
practical, 2-round solutions which are secure under this deﬁnition. Motivated by this recent work,
we introduce a deﬁnition of security against man-in-the-middle attacks that is weaker than that
considered previously [7]. In particular, we do not allow {
′
to invoke multiple concurrent executions
of { while {
′
is interacting with 1, and we do not consider reset attacks. We believe this approach
is reasonable for most network-based settings since (1) { may simply refuse to execute multiple
instances of the protocol simultaneously and (2) reset attacks are not an issue in a network-based
setting.
Deﬁnition 5.6 Let Π = (/, {, 1) be a tuple of ppt algorithms. We say Π is an identiﬁcation
scheme secure against man-in-the-middle attacks if the following conditions hold:
(Correctness) For all (pk, sk) output by /(1
k
), we have '{
sk
, 1`(pk) = 1 (where {
sk
denotes
{(sk)).
(Security) For all ppt adversaries {
′
= ({
′
1
, {
′
2
), the following is negligible (in k):
Pr